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Thermodynamics on the horizon of a flat universe at late times is studied inholographic cosmological models that assume an associated entropy on thehorizon. In such models, a $\Lambda(t)$ model similar to a time-varying$\Lambda(t)$ cosmology is favored because of the consistency of energy flowsacross the horizon. Based on this consistency, a $\Lambda(t)$ model with apower-law term proportional to $H^{\alpha}$ is formulated to systematicallyexamine the evolution of the Bekenstein--Hawking entropy. Here, $H$ is theHubble parameter and $\alpha$ is a free parameter whose value is a real number.The present model always satisfies the second law of thermodynamics on thehorizon. In particular, the universe for $\alpha <2$ tends to approachthermodynamic equilibrium-like states. Consequently, when $\alpha < 2$, themaximization of the entropy should be satisfied as well, at least in the laststage of the evolution of an expanding universe. A relaxation-like processbefore the last stage is also examined from a thermodynamics viewpoint.

Horizon thermodynamics in holographic cosmological models with a power-law term