  In this work we tie concepts derived from statistical mechanics, information
theory and contact Riemannian geometry within a single consistent formalism for
thermodynamic fluctuation theory. We derive the concrete relations
characterizing the geometry of the Thermodynamic Phase Space stemming from the
relative entropy and the Fisher-Rao information matrix. In particular, we show
that the Thermodynamic Phase Space is endowed with a natural para-contact
pseudo-Riemannian structure derived from a statistical moment expansion which
is para-Sasaki and {\eta}-Einstein. Moreover, we prove that such manifold is
locally isomorphic to the hyperbolic Heisenberg group. In this way we show that
the hyperbolic geometry and the Heisenberg commutation relations on the phase
space naturally emerge from classical statistical mechanics. Finally, we argue
on the possible implications of our results.
