Exploring thermodynamics inconsistencies in unimodular gravity: a comparative study of two energy diffusion functions

In this work we study the thermodynamics formulation for unimodular gravity under the election of two different models for the energy diffusion function. Such function encodes the current for the non-conservation of the energy-momentum tensor and is usually termed as $Q(t)$. In analogy to the cosmological scenario where the cosmic expansion is influenced by $Q(t)$, the thermodynamics implications in this scheme are also determined by the choice of the function $Q(t)$, as we discuss in the work. Specifically, we consider the barotropic and the continuous spontaneous localization models as energy diffusion functions, commonly used in the literature as viable candidates to face the well-known $H_{0}$ tension. The consistency conditions demanded for the entropy of the system in terms of the cosmological parameters of the model: positive production ($dS/dt>0$) and convexity condition ($d^{2}S/dt^{2} <0$), are investigated. We show that these conditions strongly constraint the viability of both models. Additionally, we comment about our results and compare with those obtained in recent works where the restriction of the parameters for these two diffusion models was implemented with the use of cosmological data.