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In this work we further extend the analysis of $f(R)$ theories of gravity inthe metric formalism under the approach of a Thermodynamics analogy, proposedin arXiv:1911.04830v3. Here we assume a double-well inflationary potential inthe Einstein frame and obtain a parametric form of $f(R)$ in the correspondingJordan frame.  The whole Thermodynamics picture then follows: an equation of state, binodaland spinodal curves, phase transition, critical quantities (pressure, volumeand temperature), entropy jumps, specific-heat divergence (and thecorresponding critical exponent) and a butterfly catastrophe.

Thermodynamics of $f(R)$ Gravity: The Double Well Potential Case