Entropy of cosmological black holes and generalized second law in phantom energy-dominated universe

Adopting the thin-layer improved brick-wall method, we investigate the thermodynamics of a black hole embedded in a spatially flat Friedmann-Robertson-Walker universe. We calculate the temperature and the entropy at every apparent horizon for arbitrary solution of the scale factor. We show that the temperature and entropy display a non-trivial behavior as functions of time. In the case of black holes immersed in universe driven by phantom energy, we show that for specific ranges of the equation-of-state parameter and apparent horizons the entropy is compatible with the D-bound conjecture, even the null, dominant and strong energy conditions are violated. In the case of accretion of phantom energy onto black hole with small Hawking-Hayward quasi-local mass, we obtain an equation-of-state parameter in the range $w\leq-5/3$, guaranteeing the validity of the generalized second law.