  The Hadamard renormalization prescription is used to derive a two dimensional
analog of the renormalized stress tensor for a minimally coupled scalar field
in Schwarzschild-de Sitter space time. In the two dimensional analog the
minimal coupling reduces to the conformal coupling and the stress tensor is
found to be determined by the (nonlocal) contribution of the anomalous trace
and some additional parameters in close relation to the work [1]. To properly
relate the stress tensor to the state of outwards signals coming from the
direction of the black hole at late times we propose a cut-off hypothesis which
excludes the contribution of the anomalous trace close to the black hole
horizon. The corresponding cut-off scale is found to be related to the
equilibrium-temperature of the cosmological horizon in a leading order
estimate. Finally, we establish a relation between the radiation-temperature of
the black hole horizon at large distance from the hole and the the anomalous
trace and determine the correction term to the Hawking temperature due to the
presence of the cosmological horizon.
