  We study $D$-dimensional gauge theory with an extra dimension of a circle at
finite temperature. We mainly focus on the expectation value of the gauge field
for the direction of the extra dimension, which is the order parameter of the
gauge symmetry breaking. We evaluate the effective potential in the one-loop
approximation at finite temperature. We show that the vacuum configuration of
the theory at finite temperature is determined by a $(D-1)$-dimensional gauge
theory defined by removing the Euclidean time coordinate as well as all of the
fermions from the original $D$-dimensional gauge theory on the circle. It is
pointed out that gauge symmetry nonrestoration and/or inverse symmetry breaking
can occur at high temperature in a class of gauge theories on circles and that
phase transitions (if they occur) are, in general, expected to be first order.
