  In this work we consider conformal gauge transformations of the geometric
structure of thermodynamic fluctuation theory. In particular, we show that the
Thermodynamic Phase Space is naturally endowed with a non-integrable
connection, defined by all those processes that annihilate the Gibbs 1-form,
i.e. reversible processes. Therefore the geometry of reversible processes is
invariant under re-scalings, that is, it has a conformal gauge freedom.
Interestingly, as a consequence of the non-integrability of the connection, its
curvature is not invariant under conformal gauge transformations and,
therefore, neither is the associated pseudo-Riemannian geometry. We argue that
this is not surprising, since these two objects are associated with
irreversible processes. Moreover, we provide the explicit form in which all the
elements of the geometric structure of the Thermodynamic Phase Space change
under a conformal gauge transformation. As an example, we revisit the change of
the thermodynamic representation and consider the resulting change between the
two metrics on the Thermodynamic Phase Space which induce Weinhold's energy
metric and Ruppeiner's entropy metric. As a by-product we obtain a proof of the
well-known conformal relation between Weinhold's and Ruppeiner's metrics along
the equilibrium directions. Finally, we find interesting properties of the
almost para-contact structure and of its eigenvectors which may be of physical
interest.
