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We compute the term $\sim g^3$ in the free energy for a $SU(N)$ gauge theorywith nonzero holonomy at nonzero temperature. If the holonomy is generatedkinematically by the introduction of gauge invariant sources coupled toPolyakov loops, the contribution of charged (off-diagonal) gluons to the freeenergy at order $g^3$, ${\cal F}^{\left( 3\rm{:c.g.} \right)}$, is singular:${\cal F}^{\left( 3\rm{:c.g.} \right)} \neq 0 $ without holonomy, but ${\calF}^{\left( 3 \rm{:c.g.} \right)}= 0$ when the holonomy is nonzero, eveninfinitesimally. We show that the absence of the charged gluon contribution isrequired by gauge invariance alone and is therefore a universal feature.

Conundrum for the free energy of a holonomous gluonic plasma at cubic order