  If SU(N) gauge fields live in a world with a circular extra dimension,
coupling there only to adjointly charged matter, the system possesses a global
Z(N) symmetry. If the radius is small enough such that dimensional reduction
takes place, this symmetry is spontaneously broken. It turns out that its fate
at high temperatures is not easily decided with straightforward perturbation
theory. Utilising non-perturbative lattice simulations, we demonstrate here
that the symmetry does get restored at a certain temperature T_c, both for a
3+1 and a 4+1 dimensional world (the latter with a finite cutoff). To avoid a
cosmological domain wall problem, such models would thus be allowed only if the
reheating temperature after inflation is below T_c. We also comment on the
robustness of this phenomenon with respect to small modifications of the model.
