  Recently we showed that in semiclassical 2D dilaton gravity the regularity of
a black hole horizon may be compatible with divergencies of Polyakov-Liouville
stresses on it, the temperature deviating from its Hawking value. This makes
the question about thermal properties of such solutions non-trivial. We
demonstrate that, adding to gravitation-dilaton part of the action certain
counterterms, which diverge on the horizon but are finite outside it, one may
achieve finiteness of the effective gravitation-dilaton couplings on the
horizon. This gives for the entropy S the Bekenstein - Hawking value in the
nonextreme case and S=0 in the extreme one similarly to what happens to
''standard '' black holes.
