  We show how the fully resummed thermal pressure is rendered ultraviolet
finite by standard zero-temperature renormalisation. The analysis is developed
in a 6-dimensional scalar model that mimics QED and has $N$ flavours. The
$N\to\infty$ limit of the model can be calculated completely. At a critical
temperature, one of the degrees of freedom has vanishing screening mass like
the transverse gauge bosons in four-dimensional finite-temperature perturbation
theory. The renormalised nonperturbative interaction pressure of this model is
evaluated numerically.
