First-order phase transitions and cosmic evolution: thermodynamic approach to generalized holographic dark energy

Focusing on the description of cosmic evolution at late times, this study examines a generalized holographic dark energy (HDE) framework constructed via a polynomial expansion in the Hubble parameter, which includes contributions proportional to $H^{2}$, $H^{4}$, and $H^{6}$, introduced through a variable parameter within the standard holographic formula. The analysis is carried out in the context of a spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) Universe, consisting of non-interacting matter together with the HDE fluid. We obtain the full set of Friedmann equations to investigate cosmic evolution and then analyze the system to determine whether thermodynamic $P - v$ type phase transitions can occur.