  We analyze the rather unusual properties of some exact solutions in 2D
dilaton gravity for which infinite quantum stresses on the Killing horizon can
be compatible with regularity of the geometry. In particular, the Boulware
state can support a regular horizon. We show that such solutions are contained
in some well-known exactly solvable models (for example, RST). Formally, they
appear to account for an additional coefficient $B$ in the solutions (for the
same Lagrangian which contains also ''traditional'' solutions) that gives rise
to the deviation of temperature $T$ from its Hawking value $T_{H}$. The
Lorentzian geometry, which is a self-consistent solution of the semiclassical
field equations, in such models, is smooth even at $B\neq 0$ and there is no
need to put B=0 ($T=T_{H}$) to smooth it out$.$ We show how the presence of
$B\neq 0$ affects the structure of spacetime. In contrast to ''usual'' black
holes, full fledged thermodynamic interpretation, including definite value of
entropy, can be ascribed (for a rather wide class of models) to extremal
horizons, not to nonextreme ones. We find also new exact solutions for
''usual'' black holes (with $T=T_{H}$). The properties under discussion arise
in the \QTR{it}{weak}-coupling regime of the effective constant of
dilaton-gravity interaction. Extension of features, traced in 2D models, to 4D
dilaton gravity leads, for some special models, to exceptional nonextreme black
holes having no own thermal properties.
