Evolution of thermodynamic quantities on cosmological horizon in $\mathrm{\Lambda }(t)$ model

The horizon of a flat Friedmann–Robertson–Walker (FRW) universe is considered to be dynamic when the Hubble parameter $H$ and the Hubble radius $r_H$ vary with time, unlike for de Sitter universes. To clarify the thermodynamics on a dynamic horizon, the evolution of a dynamical Kodama–Hayward temperature and Bekenstein–Hawking entropy on the horizon of a flat FRW universe is examined in a $\mathrm{\Lambda }(t)$ model similar to time-varying $\mathrm{\Lambda }(t)$ cosmologies. The $\mathrm{\Lambda }(t)$ model includes both a power-law term proportional to $H^{\alpha }$ (where $\alpha$ is a free variable) and the equation of state parameter $w$, extending a previous analysis [Phys. Rev. D 100, 123545 (2019) (arXiv:1911.08306)]. Using the present model, a matter-dominated universe ($w=0$) and a radiation-dominated universe ($w=1\mathrm{/}3$) are examined, setting $\alpha <2$. Both universes tend to approach de Sitter universes and satisfy the maximization of entropy in the last stage. The evolution of several parameters (such as the Bekenstein–Hawking entropy) is similar for both $w=0$ and $w=1\mathrm{/}3$, though the dynamical temperature $T_H$ is different. In particular, $T_H$ is found to be constant when $w=1\mathrm{/}3$ with $\alpha =1$, although $H$ and $r_H$ vary with time. To discuss this case, the specific conditions required for constant $T_H$ are examined. Applying the specific condition to the present model gives a cosmological model that can describe a universe at constant $T_H$, as if the dynamic horizon is in contact with a heat bath. The relaxation processes for the universe are also discussed.