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