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We consider the Hamiltonian mechanics and thermodynamics of an eternal blackhole in a box of fixed radius and temperature in generic 2-D dilaton gravity.Imposing boundary conditions analoguous to those used by Louko and Whiting forspherically symmetric gravity, we find that the reduced Hamiltonian genericallytakes the form: $$ H(M,\phi_+) = \sigma_0 E(M,\phi_+) -{N_0\over 2\pi} S(M) $$where $E(M,\phi_+)$ is the quasilocal energy of a black hole of mass $M$ insidea static box (surface of fixed dilaton field $\phi_+$) and $S(M)$ is theassociated classical thermodynamical entropy. $\sigma_0$ and $N_0$ determinetime evolution along the world line of the box and boosts at the bifurcationpoint, respectively. An ansatz for the quantum partition function is obtainedby fixing $\sigma_0$ and $N_0$ and then tracing the operator $e^{-\beta H}$over mass eigenstates. We analyze this partition function in some detail bothgenerically and for the class of dilaton gravity theories that is obtained bydimensional reduction of Einstein gravity in n+2 dimensions with $S^n$spherical symmetry.Comment: 25 pages Revtex including 7 (eps) figure

Hamiltonian thermodynamics of black holes in generic 2D dilaton gravity