  We extend a recent computation of the dependence of the free energy, F, on
the noncommutative scale $\theta$ to theories with very different UV
sensitivity. The temperature dependence of $F$ strongly suggests that a reduced
number of degrees of freedom contributes to the free energy in the non-planar
sector, $F_{\rm np}$, at high temperature. This phenomenon seems generic,
independent of the UV sensitivity, and can be traced to modes whose thermal
wavelengths become smaller than the noncommutativity scale. The temperature
dependence of $F_{\rm np}$ can then be calculated at high temperature using
classical statistical mechanics, without encountering a UV catastrophe even in
large number of dimensions. This result is a telltale sign of the low number of
degrees of freedom contributing to $F$ in the non-planar sector at high
temperature. Such behavior is in marked contrast to what would happen in a
field theory with a random set of higher derivative interactions.
