  The present work extends the well-known thermodynamic relation $C=\beta ^{2}<
\delta {E^{2}}>$ for the canonical ensemble. We start from the general
situation of the thermodynamic equilibrium between a large but finite system of
interest and a generalized thermostat, which we define in the course of the
paper. The resulting identity $< \delta \beta \delta {E}> =1+< \delta {E^{2}}%
> \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states
with a negative heat capacity $C<0$; at the same time, it represents a
thermodynamic fluctuation relation that imposes some restrictions on the
determination of the microcanonical caloric curve $\beta (E) =\partial S(E)
/\partial E$. Finally, we comment briefly on the implications of the present
result for the development of new Monte Carlo methods and an apparent analogy
with quantum mechanics.
