  We present an analytical formulation of the thermodynamics, free energy and
entropy, of any generic Bogoliubov de Genes model which develops exceptional
point (EP) bifurcations in its complex spectrum when coupled to reservoirs. We
apply our formalism to a non-Hermitian Josephson junction where, despite recent
claims, the supercurrent does not exhibit any divergences at EPs. The entropy,
on the contrary, shows a universal jump of $1/2\log 2$ which can be linked to
the emergence of Majorana zero modes (MZMs) at EPs. Our method allows us to
obtain precise analytical boundaries for the temperatures at which such
Majorana entropy steps appear. We propose a generalized Maxwell relation
linking supercurrents and entropy which could pave the way towards the direct
experimental observation of such steps in e.g. quantum-dot based minimal Kitaev
chains.
