  By replacing the internal energy with the free energy, as coordinates in a
"space of observables", we slightly modify (the known three) non-holonomic
geometrizations and show that the coefficients of the curvature tensor field,
of the Ricci tensor field and the scalar curvature function still remain
rational functions. In addition, we define and study a new holonomic Riemannian
geometric model associated, in a canonical way, to the Gibbs-Helmholtz equation
from Classical Thermodynamics. The main geometric invariants are determined and
some of their properties are derived. Using this geometrization, we
characterize the equivalence between the Gibbs-Helmholtz entropy and the
Boltzmann-Gibbs-Shannon, Tsallis and Kaniadakis entropies, respectively, by
means of three stochastic integral equations. We prove that some specific
(infinite) families of normal probability distributions are solutions for these
equations.
