  Expressions for the thermodynamic potential of a Dirac fermion gas are
represented at finite temperature with the chemical potential in an ultrastatic
space $R^d\times S^N$. The high- and low- temperature expansions for the
thermodynamic potential are obtained and, in particular, strongly degenerate
fermi gas is investigated. For the Candelas-Weinberg model, sufficiently high
"charge" density prevents the compactification of the extra space.
