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≠0 and there is no need to put B=0 (T=T H ) to smooth it out. We show how the presence of B≠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 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. (shrink)