  We study the covariantly constant Savvidy-type chromomagnetic vacuum in
finite-temperature Yang-Mills theory on the four-dimensional curved spacetime.
Motivated by the fact that a positive spatial curvature acts as an effective
gluon mass we consider the compact Euclidean spacetime $S^1\times S^1\times
S^2$, with the radius of the first circle determined by the temperature
$a_1=(2\pi T)^{-1}$. We show that covariantly constant Yang-Mills fields on
$S^2$ cannot be arbitrary but are rather a collection of monopole-antimonopole
pairs. We compute the heat kernels of all relevant operators exactly and show
that the gluon operator on such a background has negative modes for any compact
semi-simple gauge group. We compute the infrared regularized effective action
and apply the result for the computation of the entropy and the heat capacity
of the quark-gluon gas. We compute the heat capacity for the gauge group SU(2N)
for a field configuration of $N$ monopole-antimonopole pairs. We show that in
the high-temperature limit the heat capacity is well defined in the infrared
limit and exhibits a typical behavior of second-order phase transition $\sim
(T-T_c)^{-3/2}$ with the critical temperature $T_c=(2\pi a)^{-1}$, where $a$ is
the radius of the 2-sphere $S^2$.
