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We present a general way to define regular vacuum states of a quantized massless scalar field in two-dimensional spacetimes with horizons. We discuss in more detail the cases of Schwarzschild-de Sitter spacetime and especially the exterior of a massive shell that collapses to form a black hole in a two-dimensional de Sitter spacetime. In the latter case a vacuum state is defined using modes of positive frequency with respect to the past cosmological horizon's affine parameter. In this vacuum static observers, long after the shell has collapsed, detect thermal fluxes coming from both the cosmological and the black-hole horizons, and characterized by the corresponding Hawking temperatures. The renormalized stress-energy tensor in this vacuum state is regular everywhere and has precisely the form one would expect from prior experience with de Sitter spacetime and with gravitational collapse spacetimes that have a vanishing cosmological constant.
Marković et al. (Tue,) studied this question.