Evolution of thermodynamic quantities on cosmological horizon in $\Lambda(t)$ 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 $\Lambda(t)$ model similar to time-varying $\Lambda(t)$ cosmologies. The $\Lambda(t)$ 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 ($w=1/3$) 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 $w=1/3$, though the dynamical temperature $T_{H}$ is different. In particular, $T_{H}$ is found to be constant when $w=1/3$ 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.