  In this work we give a characterisation of first order phase transitions as
equilibrium processes on the thermodynamic phase space for which the Legendre
symmetry is broken. Furthermore, we consider generalised theories of
thermodynamics, where the potential is a homogeneous function of any order
$\beta$ and we propose a (contact) Hamiltonian formulation of equilibrium
processes. Indeed we prove that equilibrium corresponds to the zeroth levels of
such function. Using these results we infer that the description in equilibrium
of first order phase transitions is possible only when the potential is a
homogeneous function of order one, unless a generalised Zeroth Law is
postulated in order to allow for equilibrium between sub-parts of the system at
different values of the intensive quantities. Finally, we show the example of
the Tolman-Ehrenfest effect.
