  We study the equilibrium properties of the repulsive quantum Bose-Hubbard
model at high temperatures in arbitrary dimensions, with and without disorder.
In its microcanonical setting the model conserves energy and particle number.
The microcanonical dynamics is characterized by a pair of two densities: energy
density $\varepsilon$ and particle number density $n$. The macrocanonical Gibbs
distribution also depends on two parameters: the inverse nonnegative
temperature $\beta$ and the chemical potential $\mu$. We prove the existence of
non-Gibbs states, that is, pairs $(\varepsilon,n)$ which cannot be mapped onto
$(\beta,\mu)$. The separation line in the density control parameter space
between Gibbs and non-Gibbs states $\varepsilon \sim n^2$ corresponds to
infinite temperature $\beta=0$. The non-Gibbs phase cannot be cured into a
Gibbs one within the standard Gibbs formalism using negative temperatures.
