Mass function and particle creation in Schwarzschild–de Sitter spacetime

We construct a mass or energy function for the non-Nariai class Schwarzschild–de Sitter black hole spacetime in the region between the black hole and the cosmological event horizons. The mass function is local, positive definite, continuous and increases monotonically with the radial distance from the black hole event horizon. We derive the Smarr formula using this mass function, and demonstrate that the mass function reproduces the two-temperature Schwarzschild–de Sitter black hole thermodynamics, along with a term corresponding to the negative pressure due to positive cosmological constant. We further give a field theoretic derivation of the particle creation by both the horizons and discuss its connection with the mass function.