  In an approach to noncommutative gauge theories, where the full
noncommutative behavior is delimited by the presence of the UV and IR cutoffs,
we consider the possibility of describing a system at a temperature T in a box
of size L. Employing a specific form of UV/IR relationship inherent in such an
approach of restrictive noncommutativity, we derive, for a given temperature T,
an upper bound on the parameter of spacetime noncommutativity Lambda_NC ~
|theta|^{-1/2}. Considering such epochs in the very early universe which are
expected to reflect spacetime noncommutativity to a quite degree, like the
reheating stage after inflation, or believable pre-inflation
radiation-dominated epochs, the best limits on Lambda_NC are obtained. We also
demonstrate how the nature and size of the thermal system (for instance, the
Hubble distance versus the future event horizon) can affect our bounds.
