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

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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the case of black holes with non-constant scalar curvature $R$, the thermodynamical equivalence of the two frames is no longer valid.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.