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The thermodynamic behavior of the Bose–Hubbard model is solved for any temperature and any chemical potential. It is found that there is a range of critical coupling strengths λc1 < λc2 < λc3 < … in this model. For coupling strengths between λc,k and λc,k+1, Bose–Einstein condensation is suppressed at densities near the integer values ρ = 1,…,k with an energy gap. This is known as a Mott insulator phase and was previously shown only for zero temperature. In the context of ultra-cold atoms, this phenomenon was experimentally observed in 2002(1) but, in the Bose–Hubbard model, it manifests itself also in the pressure-volume diagram at high pressures. It is suggested that this phenomenon persists for finite-range hopping and might also be experimentally observable.