Black hole entropy: Off shell versus on shell

Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are compared. The general relation between on-shell and off-shell approaches is discussed. We consider the simplified 2D model with the dilaton gravity and show in its framework that the observable thermodynamical black hole entropy can be presented in the form S-TD=pi r(+)(-2)+S-SM-S-Rindler(SM). Here, (r) over bar(+) is the radius of the horizon shifted because of the quantum back reaction effect, S-SM is the statistical-mechanical entropy, determined as S-SM=-Tr(<(rho)over cap>(H)ln<(rho)over cap>(H)) for the density matrix <(rho)over cap (H)> of a black hole, and S-Rindler(SM) is the analogous entropy in the Rindler space.