  The Schwarzschild-de Sitter (SdS) metric is the simplest spacetime solution
in general relativity with both a black hole event horizon and a cosmological
event horizon. Since the Schwarzschild metric is the most simple solution of
Einstein's equations with spherical symmetry and the de Sitter metric is the
most simple solution of Einstein's equations with a positive cosmological
constant, the combination in the SdS metric defines an appropriate background
geometry for semi-classical investigation of Hawking radiation with respect to
past and future horizons. Generally, the black hole temperature is larger than
that of the cosmological horizon, so there is heat flow from the smaller black
hole horizon to the larger cosmological horizon, despite questions concerning
the definition of the relative temperature of the black hole without a
measurement by an observer sitting in an asymptotically flat spacetime. Here we
investigate the accelerating boundary correspondence (ABC) of the radiation in
SdS spacetime without such a problem. We have solved for the boundary dynamics,
energy flux and asymptotic particle spectrum. The distribution of particles is
globally non-thermal while asymptotically the radiation reaches equilibrium.
