  At a nonzero temperature T, a constant field $\overline{A}_0 \sim T/g$
generates nontrivial eigenvalues of the thermal Wilson line. We discuss
contributions to the free energy of such a holonomous plasma when the coupling
constant, $g$, is weak. We review the computation to $\sim g^2$ by several
alternate methods, and show that gauge invariant sources, which are nonlinear
in the gauge potential $A_0$, generate novel contributions to the gluon self
energy at $\sim g^2$. These ensure the gluon self energy remains transverse to
$\sim g^2$, and are essential in computing contributions to the free energy at
$\sim g^3$ for small holonomy, $\overline{A}_0 \sim T$. We show that the
contribution $\sim g^3$ from off-diagonal gluons is discontinuous as the
holonomy vanishes. The contribution from diagonal gluons is continuous as the
holonomy vanishes, but sharply constrains the possible sources which generate
nonzero holonomy, and must involve an infinite number of Polyakov loops.
