Thermodynamical correspondence of $f(R)$ gravity in 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the case of black holes with non-constant scalar curvature $R$, the thermodynamical equivalence of the two frames is no longer valid. In addition, we examine cosmological solutions with non-trivial curvatures and as we demonstrate 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.