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

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.