This paper presents an analytical formulation of the thermodynamics, free energy, and entropy of non-Hermitian Josephson junctions with exceptional points (EPs). The authors develop a method to calculate the free energy of open quantum systems without divergences at EPs, which allows for the calculation of physical observables such as entropy and supercurrents. The entropy shows a universal jump of 1/2 log 2 at EPs, which is linked to the emergence of Majorana zero modes (MZMs). The authors propose a generalized Maxwell relation connecting supercurrents and entropy, which could enable the experimental observation of these effects in quantum-dot based minimal Kitaev chains. The paper discusses the thermodynamics of non-Hermitian systems, focusing on the role of EPs in open Bogoliubov-de Gennes (BdG) models. The authors show that the entropy changes at EPs can be understood as a result of the emergence of MZMs, while the free energy remains largely insensitive to EPs. The authors also apply their formalism to a non-Hermitian Josephson junction, demonstrating that the supercurrent does not exhibit any divergences at EPs, but the entropy does show a fractional plateau. The paper also presents a detailed analysis of a non-Hermitian minimal Kitaev Josephson junction, showing how EPs can be used to calculate critical temperatures and entropy changes. The authors propose a method for experimentally detecting fractional entropy changes using the Josephson current, which could provide a signature of MZMs in the junction. The paper concludes with a discussion of the implications of their findings for the study of non-Hermitian systems and the potential for experimental verification of the predicted effects.This paper presents an analytical formulation of the thermodynamics, free energy, and entropy of non-Hermitian Josephson junctions with exceptional points (EPs). The authors develop a method to calculate the free energy of open quantum systems without divergences at EPs, which allows for the calculation of physical observables such as entropy and supercurrents. The entropy shows a universal jump of 1/2 log 2 at EPs, which is linked to the emergence of Majorana zero modes (MZMs). The authors propose a generalized Maxwell relation connecting supercurrents and entropy, which could enable the experimental observation of these effects in quantum-dot based minimal Kitaev chains. The paper discusses the thermodynamics of non-Hermitian systems, focusing on the role of EPs in open Bogoliubov-de Gennes (BdG) models. The authors show that the entropy changes at EPs can be understood as a result of the emergence of MZMs, while the free energy remains largely insensitive to EPs. The authors also apply their formalism to a non-Hermitian Josephson junction, demonstrating that the supercurrent does not exhibit any divergences at EPs, but the entropy does show a fractional plateau. The paper also presents a detailed analysis of a non-Hermitian minimal Kitaev Josephson junction, showing how EPs can be used to calculate critical temperatures and entropy changes. The authors propose a method for experimentally detecting fractional entropy changes using the Josephson current, which could provide a signature of MZMs in the junction. The paper concludes with a discussion of the implications of their findings for the study of non-Hermitian systems and the potential for experimental verification of the predicted effects.