  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curvature $R$, the thermodynamical equivalence of the two frames is no longer
valid.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.
