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Thermodynamics of the FRW Universe in Generalized Proca Theory

We investigate the thermodynamics of a spatially flat Friedmann-Robertson-Walker (FRW) universe within the framework of Generalized Proca (GP) theory, a comprehensive vector-tensor theory. By adopting two distinct dark energy models in GP, we derive the corresponding modified Friedmann equations, the thermodynamic first law for the apparent horizon, and the equation of state for these models. For the first model, characterized by power-law couplings and an ansatz linking the Proca field to the Hubble parameter, we analytically demonstrate the existence of a critical point in the pressure-volume$(P-V)$ diagram, indicating a $P-V$ phase transition. For the second model, defined by a Proca field marginal coupled to curvature, we show that there is no phase transition when the coupling constants are selected within the range permitted by observations. This investigation not only extends the FRW thermodynamics to vector-tensor theories but also demonstrates that cosmological phase transitions can serve as a powerful diagnostic tool for distinguishing viable dark energy models in GP theory.