The entropy evolution of a Schwarzschild-(Anti) de Sitter black hole

Based on the definition of the interior volume of spherically symmetry black holes, the interior volume of Schwarzschild-(Anti) de Sitter black holes is calculated. It is shown that with the cosmological constant (A) increasing, the changing behaviors of both the position of the largest hypersurface and the interior volume for the Schwarzschild-Anti de Sitter black hole are the same as the Schwarzschild-de Sitter black hole. Considering a scalar field in the interior volume and Hawking radiation with only energy, the evolution relation between the scalar field entropy and Bekenstein-Hawking entropy is constructed. The results show that the scalar field entropy is approximately proportional to Bekenstein-Hawking entropy during Hawking radiation. Meanwhile, the proportionality coefficient is also regarded as a constant approximately with the increasing A. Furthermore, considering A as a dynamical variable, the modified Stefan-Boltzmann law is proposed which can be used to describe the variation of both the mass and A under Hawking radiation. Using this modified law, the evolution relation between the two types of entropy is also constructed. The results show that the coefficient for Schwarzschild-de Sitter black holes is closer to a constant than the one for Schwarzschild-Anti de Sitter black holes during the evaporation process. Moreover, we find that for Hawking radiation carrying only energy, the evolution relation is a special case compared with the situation that the mass and A are both considered as dynamical variables.