Backreaction issue for the black hole in de Sitter spacetime

We consider a quantum real massive scalar field in the de Sitter-Schwarzschild spacetime background. To have an analytic headway we study in detail the two-dimensional case, assuming that the situation in four dimensions will not be much different conceptually. It is assumed that the quantum field is in a thermal state, i.e., described by the Planckian distribution for the exact modes in the geometry under consideration. We calculate approximately the expectation value of the stress-energy tensor near the cosmological and black hole horizons. It is shown that for a generic temperature backreaction from quantum fields, the geometry cannot be neglected. Thus, de Sitter-Schwarzschild spacetime geometry inevitably is strongly modified by the quantum fluctuations of the matter fields.