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