  The first quantum correction to the finite temperature partition function for
a self-interacting massless scalar field on a $D-$dimensional flat manifold
with $p$ non-commutative extra dimensions is evaluated by means of dimensional
regularization, suplemented with zeta-function techniques. It is found that the
zeta function associated with the effective one-loop operator may be nonregular
at the origin. The important issue of the determination of the regularized
vacuum energy, namely the first quantum correction to the energy in such case
is discussed.
