  We investigate thermodynamic properties of lattice Bose gases in optical
cavities in the Mott-insulator limit. We find the system assumes anomalous
thermodynamic behavior that can be traced back to the breaking of fundamental
additivity by its infinite-long range interaction. Specifically, the system
shows striking ensemble inequivalence between the canonical ensemble and the
grand canonical one, sharply manifesting in the distinct anomalous structure of
the thermodynamic phase diagram in the canonical ensemble. In particular, in
the temperature regime around half of the on-site energy, the system manifests
negative compressibility and anomalous reentrant phase transitions where the
ordered charge density wave phase revives from the disordered homogenous phase
upon increasing the temperature. Direct experimental observation of the
anomalous behavior can be realized in the current experiments with
well-controlled total particle number fluctuations.
