==== Front Entropy (Basel) Entropy (Basel) entropy Entropy 1099-4300 MDPI 33287015 10.3390/e22111246 entropy-22-01246 Article Horizon Thermodynamics in D-Dimensional f(R) Black Hole Zhu Chenrui 1† https://orcid.org/0000-0001-6798-2190Yang Rong-Jia 123*† 1 College of Physical Science and Technology, Hebei University, Baoding 071002, China; zhuchenrui9@163.com 2 Hebei Key Lab of Optic-Electronic Information and Materials, Hebei University, Baoding 071002, China 3 Key Laboratory of High-pricision Computation and Application of Quantum Field Theory of Hebei Province, Hebei University, Baoding 071002, China * Correspondence: yangrongjia@tsinghua.org.cn† These authors contributed equally to this work. 02 11 2020 11 2020 22 11 124616 9 2020 22 10 2020 © 2020 by the authors.2020Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).We consider whether the new horizon-first law works in higher-dimensional f(R) theory. We firstly obtain the general formulas to calculate the entropy and the energy of a general spherically-symmetric black hole in D-dimensional f(R) theory. For applications, we compute the entropies and the energies of some black hokes in some interesting higher-dimensional f(R) theories. horizon thermodynamicsentropyenergyblack holef(R) theory ==== Body 1. Introduction Since Bekenstein’s and Hawking’s work [1,2], it is convinced that there may be a deep relation between the gravitational field equations and the laws of thermodynamics. Like in thermodynamics, four laws of black hole dynamics were found in [3]. The field equations of general relativity in its tensorial form can be derived by applying the Clausius relation δQ=TδS on the horizon of spacetime, here δQ is the energy flux across the horizon and δS and T are the change in the entropy and the Unruh temperature seen by an accelerating observer just inside the horizon [4]. From Einstein equations, one can obtain the thermal entropy density of spacetime without assuming the temperature or the horizon [5,6]. It was shown that for a generalized gravity theory, the field equations are equivalent to the first law of thermodynamics [7]. This programme was also applied to other modified gravity theories: such as f(R) theory [8,9], and scalar-Gauss–Bonnet gravity [10]. It was shown, however, that the Bekenstein–Hawking entropy depends not only on the black hole parameter, but also on the coupling which induces Lorentz violation [11]. For a spherically-symmetric spacetime, Einstein’s field equations can be written in the form of thermodynamic identity (called the horizon-first law): dE=TdS−PdV [12]. This framework of horizon thermodynamics has also been extended to other theories of gravity [13,14] and the non-spherically-symmetric cases [15]. The horizon-first law, however, has two shortcomings: (a) the thermodynamic variables are vague in the original derivation and require further determination, (b) both S and V are functions of only r+, so does the horizon-first law, which makes the terms ‘heat’ and ‘work’ confused [16]. To avoid these two problems, a new horizon-first law was proposed in [16], where the temperature T and the pressure P are independent thermodynamic quantities and the entropy and free energy are derived concepts, while the horizon-first law can be restored by the Legendre projection. This procedure was generalized to in f(R,RμνRμν) theory [17] and f(R) theory with a spherically-symmetric black hole [18] or with a general spherically-symmetric black hole [19]. The natural generalization of general relativity is the higher-dimensional and higher-order gravity. Higher-dimensional black hole in higher-dimensional gravity is physically interesting, whose physics are markedly different and much richer than those in four dimensions, see, for example, there are limits on the ratio of mass to charge for Tangherlini–Reissner–Nordstrom black hole [20]. Extra dimensions are also needed for consistency in string theory. So it is valuable to study the physics of high-dimensional black holes. In addition, energy issue in higher-dimensional and higher-order gravity is still an open problem, some attempts to find a satisfactory answer to this problem have been proposed [21,22,23,24,25]. In literature, there have been several attempts to define the concept of energy using local or quasi-local concepts, however, not all these definitions of energy agree with each other. Here we will investigate this issue in D-dimensional f(R) black hole and hope to give interesting suggestions. As one of the simplest modifications to general relativity, f(R) gravity have been extensively studied over the past decade [26,27,28]. It’s Lagrangian is a function of Ricci scalar in which higher-order terms can encapsulate high-energy modifications to general relativity, but the field equations are simple enough that it is possible to solve them. Secondly, and most importantly, f(R) gravity does not suffer from Oströgradsky instability. Various applications of f(R) gravity to cosmology have been investigated, such as inflation, dark energy, cosmological perturbations, and black hole solutions. Here we will investigate whether the new horizon-first law still works in higher-dimensional f(R) gravity. We will adopt the method presented in [12] to define the energy and the entropy: writing the radial component of gravitational field equations on the horizon as the equation of state P=D(r+)+C(r+)T, which can be rewritten as a thermodynamic identity δG=−SδT+VδP, then identifying S as the entropy, and taking E=G+TS−PV as the energy. We will show that the new horizon-first law can give not only the entropy but also the energy of black hole in higher-dimensional f(R) theory, which for some special cases are consistent with the results obtained by using other methods. The structure of this paper is as follows. In Section 2, we briefly review the new horizon-first law and its applications in f(R) theories. In Section 3, we consider whether the new horizon-first law still holds in higher-dimensional f(R) theory. In Section 4, we discuss applications for some D-dimensional f(R) theories. Conclusions and discussions are given in Section 5. 2. The New Horizon-First Law and Its Application in f(R) Theory Inspired by the radial Einstein equation on the horizon of Schwarzschild black hole, it is reasonable to suggest that the radial field equation of a gravitational theory under consideration takes the following form [16] (1) P=D(r+)+C(r+)T, where C and D are functions of the radius of black hole, r+, in general they depend on the gravitational theory one considered. The temperature T in (1) is identified from thermal quantum field theory, which is independent of any gravitational field equations [16]. According to the conjecture proposed in [6], the pressure in (1) is identified as the (rr) component of the matter stress-energy, it also does not fall back on any gravitational field equations. Considering a virtual displacements δr+ and varying the Equation (1), then multiplying the volume of black hole V(r+), yields [16] (2) δG=−SδT+VδP, comparing with the thermodynamical identity δG=−SδT+VδP, where G can be identified as the Gibbs free energy which is given by [16] (3) G=∫V(r+)D′(r+)dr++T∫V(r+)C′(r+)dr+=PV−ST−∫V′(r+)D(r+)dr+, and S is identified as the entropy which is [16] (4) S=∫V′(r+)C(r+)dr+. Using the degenerate Legendre transformation as that in thermodynamics, the energy E is defined as E=G+TS−PV which can be easily got [18] (5) E=−∫V′(r+)D(r+)dr+. This procedure was firstly investigated for Einstein gravity and Lovelock gravity which only give rise to second-order field equation [16] and was also applied to f(R) gravity with a general static spherically-symmetric black hole in f(R) gravity (6) ds2=−W(r)dt2+dr2N(r)+r2dΩ2, where W(r) and N(r) are general functions of the coordinate r and the event horizon is local at the largest positive root of N(r+)=0 with N′(r+)≠0, the entropy in this case is given by [19] (7) S=∫(2πr+F+πr+2F′)dr+=πr+2F, where F=dfdR. The energy is found to be [19] (8) E=12∫W′N′Fr+2+12(f−RF)r+2dr+. For W(r)=N(r), Equation (8) reduces to the result obtained in [18] which is consistent with the expression obtained in [21] and can be derived by using the unified first law of black hole dynamics [29]; Equation (7) is consistent with the results derived by using the Wald entropy formula or the Euclidean semiclassical approach [30,31,32]. In the next section, we will consider the new horizon-first law in D-dimensional f(R) Theory with a general static spherically-symmetric black hole. 3. The Entropy and Energy of D-Dimensional f(R) Black Hole In this section, we turn our attention to discussing whether the new horizon-first law still holds in the D-dimensional f(R) theory. Considering a general spherically-symmetric and static D-dimensional black hole in f(R) theory, its geometry is given by (9) ds2=−W(r)dt2+dr2N(r)+r2dΩD−22, in which dΩD−22 represents the D−2-dimensional unit spherical line element. For the metric (9), the surface gravity takes the form [33]: κK=W′(r+)N′(r+)/2, giving the temperature of the black hole as (10) T=κK2π=W′(r+)N′(r+)4π. The action of D-dimensional f(R) gravity with source is represented by (11) I=∫dDx−gf(R)2k2+Lm, where k2=8π and D≥3. Here we take the units G=c=ℏ=1. f(R) is a function of the Ricci scalar R and Lm is the matter Lagrangian. Physically f(R) theory must fulfil two stability conditions [34]: (a) no ghosts, df/dR>0; and (b) no tachyons, d2f/dR2>0 [35]. Variation of the action (11) with respect to metric provides the gravitational field equations (12) Gμν≡Rμν−12δμνR=k21FTμν+1k2Tμν, where Tμν=−2−gδLmδgμν the energy-momentum tensor of the matter. We define the stress-energy tensor of the effective curvature fluid as Tμν which is given by (13) Tμν=1F(R)12δμν(f−RF)+∇μ∇νF−δμν□F, where □=∇λ∇λ. Assuming the metric (9), we derive after some calculations by using of the relations □F=1−g∂μ[−ggμν∂νF] the (11) components of the Einstein tensor and the effective curvature fluid respectively (14) G11=1r2(D−2)(D−3)(N−1)2+(D−2)NW′r2W, and (15) T11=1F(R)12(f−RF)−N2WW′F′−D−2rNF′, where the prime stands for the derivative with respected to r. Taking the trace of Equation (12), yields the relation (16) RF(R)−D2f(R)+(D−1)□F=k2Tνν, where Tνν is the trace of the energy-momentum tensor. Substituting Equations (14), (15) and Trr=P into Equation (12), yields (17) k2P=F(D−2)(D−3)2r2(N−1)−12(f−RF)+NF′(D−2)r+(D−2)NW′F2Wr+NW′F′2W. Thinking of N(r+)=0 and the temperature (10) at the horizon, Equation (17) reduces to (18) P=−18πF(D−3)(D−2)2r+2+12f−RF+14N′W′F(D−2)r++F′T. Comparing Equations (18) and (1), we then get (19) D(r+)=−18πF(D−3)(D−2)2r+2+12f−RF, and (20) C(r+)=14N′W′F(D−2)r++F′. The volume V of the black hole in D-dimensional spacetime takes the form [36] (21) V(r+)=2πD−12Γ(D−12)∫0r+W(r)N(r)rD−2dr. Use the relation N(r+)W(r+)=N′(r+)W′(r+) [33], we get (22) V′(r+)=2πD−12Γ(D−12)W′(r+)N′(r+)r+D−2. Substituting Equations (22) and (20) into the expression (4), the entropy of black hole (9) in D-dimensional f(R) gravity is (23) S=πD−122Γ(D−12)∫[F(D−2)r+D−3+r+D−2F′]dr+=πD−122Γ(D−12)Fr+D−2. Inserting Equations (22) and (19) into Equation (5), then we obtain the energy of black hole (9) in D-dimensional f(R) theory as (24) E=πD−324Γ(D−12)∫W′N′F(D−2)(D−3)2r+2+12(f−RF)r+D−2dr+. Equations (23) and (24) are the main results obtained in this work, they can be used to calculate the entropy and the energy of a specific black hole in a specific D-dimensional f(R) gravity. For D=4, Equations (23) and (24) recover Equations (7) and (8). Fixing f(R)=R and W=N, one expects the results to go back to the framework of higher-dimensional Einstein’s gravity, see Equations (27) and (29) in the next section. 4. Applications In this section, we will illustrate the procedure to calculate the entropy and the energy for black holes in a certain f(R) theory by using Equations (23) and (24). These models have solutions with constant Ricci curvature (such as a Schwarzschild or a Schwarzschild-de Sitter solution) or solutions with non-constant Ricci curvature. 4.1. The Constant Ricci Curvature Case We start with the simplest but important case, F=1, which implies f=R−2Λ where −2Λ is an integration constant to be regarded as the cosmological constant. This model has a Schwarzschild or a Schwarzschild-de/anti de Sitter black hole solution [37,38] (25) W(r)=N(r)=−M−Λr2,D=3,1−2M(D−3)rD−3−2Λ(D−2)(D−1)r2,D>3, where M is the mass of black hole. The solution is Schwarzschild-de/anti de Sitter black hole solution for D>3 and it is the non-rotating BTZ black hole for D=3. The constant curvature R0 from Equation (25) is given by (26) R0=2DΛD−2. From Equation (23), the entropy is found to be (27) S=πD−122Γ(D−12)r+D−2. It reduces to S=πr+/2 for a non-rotating BTZ black hole and returns to the standard results for D=4. The energy of the black hole is obtained from Equation (24) as (28) E=−18Λr+2=18M, where −Λr+2=M for D=3 has been used at the horizon. For M=0, it reduces to results presented in [12]. For D>3, it reads (29) E=πD−324Γ(D−12)(D−2)r+D−32−Λr+D−1D−1=πD−324Γ(D−12)D−2D−3M, where we used N(r+)=0 for D>3 at the horizon. For 4-Dimensional Einstein’s gravity, Equations (27) and (29) give S=πr+2=A/4 and E=M, respectively. The nonnegativity of the energy, gives new constrains on the parameter: (D−1)(D−2)2>Λr+2 for D>3. 4.2. The Non-Constant Ricci Curvature Case We apply the same procedure for black hole solutions with non-constant curvature which are more interesting. We consider two types of f(R) theories: (a) F is a linear function of r, and (b) F is a power law function of r. 4.2.1. F(r)=1+αr In this case, F is a linear function of r with α a non-zero constant. W(r) and N(r) in three-dimensional spacetime are given by [37] (30) W(r)=N(r)=C2r2+C1αr−12−C1α2r2ln1+1αr. Function f(R(r)) reads (31) f=−4C2+4C1α2ln1+1αr−2C1α(1+2αr)r(1+αr), where Ci are integration constants with C1 related to the mass of the central object and C2 identified as the cosmological constant. The Ricci scalar R evolves as (32) R=−6C2+6C1α2ln1+1αr−C1α(2+9αr+6α2r2)r(1+αr)2. Then, the entropy formula (23) gives (33) S=πr+2(1+αr+), gives limit on parameter: α≥−1/r+ from S≥0. The energy of the black hole is obtained from Equation (24) as (34) E=r+28C2+2α2C1+2αC2r+−C1α2(1+2αr+)ln1+1αr+=C116, where N(r+)=0 was used. E≥0 gives a new constraint on the parameter: C1≥0. For D=4 spacetime, W(r) and N(r) take the forms (35) W(r)=N(r)=C2r2+12+13αr+C1r3αr−2−6α2r2+6α3r3ln1+1αr, where C2 is related to the cosmological constant. We note that Equation (35) is different from Equation (27) in [37]. Function f(R(r)) is given by (36) f=−6C2−36C1α3ln1+1αr+6αC1(−1+6α2r2+3αr)r2(1+αr)+1+2αrr2, with the Ricci scalar (37) R=−12C2−72C1α3ln1+1αr+6αC1(−1+6α2r2+6αr)(1+2αr)r2(1+αr)2+1r2. From Equation (23), the entropy of the black hole reads (38) S=πr+2(1+αr+). The energy of the black hole is obtained from Equation (24) as (39) E=r+2+r+28(−6C1α2−36α3C1r++4C2r++3α+6αC2r+2)+32α3C1r+3(2+3αr+)ln1+1αr+=C1−16α, where we used N(r+)=0. For C1=0, Equation (39) reduces to the result in [18]. To guarantee the nonnegativity of the entropy and the energy, we must have new constraints on the parameters: α≥−1/r+ and C1≥1/6α. 4.2.2. F=αra We now consider a power-law form for F(r), i.e., F=αra, with constants a and α. In this case, the W(r) and N(r) in (9) were found to be [37] (40) W=r2a(a−1)a+D−2N, and (41) N=C1r−2a2−6a+6+(2a−5)D+D2a+D−2+C2r2(D−2+2a−a2)a+D−2+(D−3)(a+D−2)2[2a2−6a+6+(2a−5)D+D2](D−2+2a−a2), where C1 and C2 are the integration constants. It returns to the Schwarzschild-de/anti de Sitter solutions for a=0 and α=1. Function f(R(r)) and the Ricci scalar R take the forms, respectively (42) f=2αC2(D−1)(a−1)(D−2+2a)ra(D−a)a+D−2a+D−2+2aα(D−1)(D−3)ra−2D−2+2a−a2, (43) R=−C2(D−1)(D−a)(D−2+2a)(a+D−2)r2a(a−1)a+D−2+a(D−1)(D−3)(a−2)(D−2+2a−a2)r2. Note that although α and a are two arbitrary constants, a must satisfy a≠2−D,1±D−1. From Equation (23) the entropy for this type black hole is (44) S=απD−122Γ(D−12)r+a+D−2. The energy of the black hole is obtained from Equation (24) as (45) E=πD−324Γ(D−12)a1r+D2−2a−3D+2aD+2a+D−2+a2r+2a2+2aD+D2−6a−5D+6a+D−22(2−2a+a2−D)[6+2a2+2a(D−3)−5D+D2], where (46) a1=αC2(a+D−2)(2−2a+a2−D)[6+2a2+2a(D−3)−5D+D2], and (47) a2=−α(a+D−2)(D−3)[(3−2D)a2+(6D−8)a+(D−2)2]. If taking a=0 and α=1, it is back to Einstein’s gravity and Equation (44) reduces to Equation (27); when D=3, C2=−Λ, and C1=−M, Equation (45) reduces to Equation (28); when D≥4, C2=−2Λ(D−1)(D−2), and C1=2M3−D, Equation (45) returns to Equation (29). For D=3, the solution becomes rather specific since the last term in (41) vanishes for all values of a. The function f(R) reads [37] (48) f(R)=a3R3−a2(1−a), with a3=4αC2(2a2−a−1)[2C2(2a+1)(a−3)]a−32(1−a)(a+1)1+a2(1−a) and a≠0. f(R) is a constant for a=3 and it is un-physical for a=1. The entropy (44) and the energy (45) respectively reduce to (49) S=α2πr+a+1=α2π−C1C2(a+1)24a+2, and (50) E=αC2(a+1)8r+4a+2a+1=−αC1(a+1)8, with r+=(−C1C2)a+14a+2. The nonnegativity of the entropy gives constraints on the parameters: C1(a+1)≤0, and S≥0 gives α≥0. For a=1/3, we have f∼R2, S=α2πr+4/3=α2π−C1C2815 and E=−αC16. For D≥4 and C2=0, the function f(R) takes the form [37] (51) f(R)=a4R1−a2, where a4=2α(a−2)a2−1a(D−1)(D−3)D−2+2a−a2a2. The entropy (44) and the energy (45) respectively reads (52) S=απD−122Γ(D−12)r+a+D−2=απD−122Γ(D−12)−C1(2a2−6a+6+(2a−5)D+D2)(D−a2+2a−2)(D−3)(D−2+a)2(D−2+a)22a2−6a+6+(2a−5)D+D2, (53) E=πD−324Γ(D−12)a2r+2a2+2aD+D2−6a−5D+6a+D−22(2−2a+a2−D)[6+2a2+2a(D−3)−5D+D2]=πD−32αC18Γ(D−12)(3−2D)a2+(6D−8)a+(D−2)22−a−D, with r+=−(D−3)(D−2+a)2C1[2a2−6a+6+(2a−5)D+D2](D−a2+2a−2)−D−2+a2a2−6a+6+(2a−5)D+D2. For the case of α=1 and a=0 the theory returns to D-dimensional Einstein’s gravity: r+=(−C1)1D−3, S=πD−122Γ(D−12)(−C1)D−2D−3, and E=πD−328Γ(D−12)(D−2)r+D−3=−πD−32(D−2)C18Γ(D−12). For a=−2, we get f∼R2, S=απD−122Γ(D−12)−C1(D2−9D+26)(D−10)(D−3)(D−4)2(D−4)2D2−9D+26, and E=πD−32αC18Γ(D−12)D2−24D+324−D, obviously D≠4,10. The nonnegativity of the entropy and the energy give new constraints on the parameters: α≥0 and C1(D2−24D+32)≤0. 5. Discussion and Conclusions We have discussed whether the new horizon-first law still holds in higher-dimensional f(R) gravity. We have derived the general formulas to calculate the entropy and the energy of a general spherically-symmetric and static D-dimensional black hole in f(R) theories, which can be obtained by using other methods. It gives a new method to rapidly compute the entropy and the energy of the black hole in f(R) theory. For applications, we have calculated the entropy and the energy of some black holes with constant Ricci curvature or with non-constant Ricci curvature in some interesting f(R) theory by using these formulas, the nonnegativity of the entropy and the energy give new constraints on the parameters. Except for the case discussed in [39] where F(R)=0, it is valuable to apply this procedure to other modified gravitational theories. Acknowledgments We thank Jing Zhai for helpful advice. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Author Contributions Conceptualization, methodology, writing—review and editing, supervision, project administration, and funding acquisition, R.-J.Y.; calculation, writing—original draft preparation, C.Z. All authors have read and agreed to the published version of the manuscript. Funding This study is supported in part by National Natural Science Foundation of China (Grant No. 11273010), Hebei Provincial Natural Science Foundation of China (Grant No. A2014201068), the Outstanding Youth Fund of Hebei University (No. 2012JQ02), and the Midwest universities comprehensive strength promotion project. Conflicts of Interest The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. ==== Refs References 1. Bekenstein J.D. Black holes and entropy Phys. Rev. D 1973 7 2333 2346 10.1103/PhysRevD.7.2333 2. Hawking S.W. Particle Creation by Black Holes Commun. Math. Phys. 1975 43 199 220 10.1007/BF02345020 3. Bardeen J.M. Carter B. Hawking S.W. The four laws of black hole mechanics Commun. Math. Phys. 1973 31 161 170 10.1007/BF01645742 4. Jacobson T. Thermodynamics of space-time: The Einstein equation of state Phys. Rev. Lett. 1995 75 1260 1263 10.1103/PhysRevLett.75.1260 10060248 5. Yang R. The thermal entropy density of spacetime Entropy 2013 15 156 10.3390/e15010156 6. Yang R.-J. Is gravity entropic force? Entropy 2014 16 4483 4488 10.3390/e16084483 7. Brustein R. Hadad M. The Einstein equations for generalized theories of gravity and the thermodynamic relation δQ =TδS are equivalent Phys. Rev. Lett. 2009 103 101301 10.1103/PhysRevLett.103.101301 19792292 8. Eling C. Guedens R. Jacobson T. Non-equilibrium thermodynamics of spacetime Phys. Rev. Lett. 2006 96 121301 10.1103/PhysRevLett.96.121301 16605892 9. Elizalde E. Silva P.J. F (R ) gravity equation of state Phys. Rev. D 2008 78 061501 10.1103/PhysRevD.78.061501 10. Bamba K. Geng C.-Q. Nojiri S. Odintsov S.D. Equivalence of modified gravity equation to the Clausius relation Europhys. Lett. 2010 89 50003 10.1209/0295-5075/89/50003 11. Chen S. Jing J. Liao H. Black hole entropy arising from massless scalar field with Lorentz violation induced by the coupling to Einstein tensor Phys. Lett. B 2015 751 474 478 10.1016/j.physletb.2015.10.087 12. Padmanabhan T. Classical and quantum thermodynamics of horizons in spherically symmetric space-times Class. Quant. Gravity 2002 19 5387 5408 10.1088/0264-9381/19/21/306 13. Paranjape A. Sarkar S. Padmanabhan T. Thermodynamic route to field equations in Lancos-Lovelock gravity Phys. Rev. D 2006 74 104015 10.1103/PhysRevD.74.104015 14. Sheykhi A. Dehghani M.H. Dehghani R. Horizon Thermodynamics and Gravitational Field Equations in Quasi Topological Gravity Gen. Rel. Gravity 2014 46 1679 10.1007/s10714-014-1679-1 15. Kothawala D. Sarkar S. Padmanabhan T. Einstein’s equations as a thermodynamic identity: The Cases of stationary axisymmetric horizons and evolving spherically symmetric horizons Phys. Lett. B 2007 652 338 342 10.1016/j.physletb.2007.07.021 16. Hansen D. Kubiznak D. Mann R. Horizon Thermodynamics from Einstein’s Equation of State Phys. Lett. B 2017 771 277 280 10.1016/j.physletb.2017.04.076 17. Feng H. Yang R.-J. Horizon thermodynamics in f (R ,Rμν Rμν ) Theory Chin. Phys. C 2020 in press 18. Zheng Y. Yang R.-J. Horizon thermodynamics in f (R ) theory Eur. Phys. J. C 2018 78 682 10.1140/epjc/s10052-018-6167-4 19. Zheng Y. Yang R.-J. Entropy and Energy of Static Spherically Symmetric Black Hole in f (R ) theory Universe 2020 6 47 10.3390/universe6030047 20. Yang R.-J. Constraints from accretion onto a Tangherlini-Reissner-Nordstrom black hole Eur. Phys. J. C 2019 79 367 10.1140/epjc/s10052-019-6886-1 21. Cognola G. Gorbunova O. Sebastiani L. Zerbini S. On the Energy Issue for a Class of Modified Higher Order Gravity Black Hole Solutions Phys. Rev. D 2011 84 023515 10.1103/PhysRevD.84.023515 22. Deser S. Tekin B. Energy in generic higher curvature gravity theories Phys. Rev. D 2003 6 084009 10.1103/PhysRevD.67.084009 23. Deser S. Tekin B. New energy definition for higher curvature gravities Phys. Rev. D 2007 75 084032 10.1103/PhysRevD.75.084032 24. Abreu G. Visser M. Tolman mass, generalized surface gravity, and entropy bounds Phys. Rev. Lett. 2010 105 041302 10.1103/PhysRevLett.105.041302 20867834 25. Cai R.-G. Cao L.-M. Hu Y.-P. Ohta N. Generalized Misner-Sharp Energy in f (R ) Gravity Phys. Rev. D 2009 80 104016 10.1103/PhysRevD.80.104016 26. De Felice A. Tsujikawa S. f (R ) theories Living Rev. Rel. 2010 13 3 10.12942/lrr-2010-3 28179828 27. Sotiriou T.P. Faraoni V. f (R ) theories of Gravity Rev. Mod. Phys. 2010 82 451 497 10.1103/RevModPhys.82.451 28. Capozziello S. De Laurentis M. Extended Theories of Gravity Phys. Rep. 2011 509 167 321 10.1016/j.physrep.2011.09.003 29. Hayward S.A. Unified first law of black hole dynamics and relativistic thermodynamics Class. Quant. Gravity 1998 15 3147 3162 10.1088/0264-9381/15/10/017 30. Dyer E. Hinterbichler K. Boundary Terms, Variational Principles and Higher Derivative Modified Gravity Phys. Rev. D 2009 79 024028 10.1103/PhysRevD.79.024028 31. Vollick D.N. Noether Charge and Black Hole Entropy in Modified Theories of Gravity Phys. Rev. D 2007 76 124001 10.1103/PhysRevD.76.124001 32. Iyer V. Wald R.M. A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes Phys. Rev. D 1995 52 4430 4439 10.1103/PhysRevD.52.4430 10019667 33. Di Criscienzo R. Hayward S.A. Nadalini M. Vanzo L. Zerbini S. Hamilton-Jacobi tunneling method for dynamical horizons in different coordinate gauges Class. Quant. Gravity 2010 27 015006 10.1088/0264-9381/27/1/015006 34. Pogosian L. Silvestri A. The pattern of growth in viable f (R ) cosmologies Phys. Rev. D 2008 77 023503 10.1103/PhysRevD.77.023503 35. Dolgov A.D. Kawasaki M. Can modified gravity explain accelerated cosmic expansion? Phys. Lett. B 2003 573 1 4 10.1016/j.physletb.2003.08.039 36. Parikh M.K. The Volume of black holes Phys. Rev. D 2006 73 124021 10.1103/PhysRevD.73.124021 37. Amirabi Z. Halilsoy M. Habib Mazharimousavi S. Generation of spherically symmetric metrics in f (R ) gravity Eur. Phys. J. C 2016 76 338 10.1140/epjc/s10052-016-4164-z 38. Bueno P. Cano P.A. On black holes in higher-derivative gravities Class. Quant. Gravity 2017 34 175008 10.1088/1361-6382/aa8056 39. Nashed G.G.L. Capozziello S. Charged spherically symmetric black holes in f (R ) gravity and their stability analysis Phys. Rev. D 2019 99 104018 10.1103/PhysRevD.99.104018