  The vacuum energy is calculated for Yang-Mills (YM) system defined in $D$
dimensional space-time of $S^1\times R^d$ ($D=d+1$), where the possibility of
the YM fields to acquire the vacuum expectation values on $S^1$ is taken into
account. The vacuum energy has already been obtained to the order of one-loop
in many people. Here we calculate the vacuum energy in $D$ dimensions to
two-loop order. With an intention to reach higher loops, an approximation
method is proposed, which is especially effective in higher dimensions. By this
method, we can treat the higher-loop contributions of YM interactions as easily
as we treat one-loop effect. As a check, we show reproduction of the two-loop
contribution ($D$-dependence of the coefficient as well as the functional form)
when the coupling constant is small. This approximation method is useful not
only for the Kaluza-Klein theories but also for the finite temperature-density
system (as a quark-gluon plasma).
