Thermodynamical correspondence of f(R) gravity in the Jordan and Einstein frames

We study the thermodynamical aspects of f (R) gravity in the Jordan and the Einstein frame, and we investigate the corresponding equivalence of the thermodynamical quantities in the two frames. We examine static spherically symmetric black hole solutions with constant Ricci scalar curvature R, and as we demonstrate, the thermodynamical quantities in the two frames are equivalent. However, for the case of black holes with nonconstant scalar curvature R, the thermodynamical equivalence of the two frames is no longer valid. in addition, we extend our study to investigate cosmological solutions with a homogeneous and isotropic background. In particular, we find that the power-law cosmology case provides an accidentally thermodynamical equivalence of the two frames. However, for nontrivial cosmology, we examine a novel exponential ultraviolet f(R) gravity. This confirms that the thermodynamical quantities in both frames are not equivalent. In conclusion, although f (R) gravity and its corresponding scalar-tensor theory are mathematically equivalent, at least for conformal invariant quantities, the two frames are not thermodynamically equivalent at a quantitative level, in terms of several physical quantities.