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We investigate the perturbative expansion in $SU(3)$ Yang-Mills theorycompactified on $\mathbb{R}^2\times \mathbb{T}^2$ where the compact space is atorus $\mathbb{T}^2= S^1_{\beta}\times S^1_{L}$, with $S^1_{\beta}$ being athermal circle with period $\beta=1/T$ ($T$ is the temperature) while $S^1_L$is a circle with finite length $L=1/M$, where $M$ is an energy scale. ALinde-type analysis indicates that perturbative calculations for the pressurein this theory break down already at order $\mathcal{O}(g^2)$ due to thepresence of a non-perturbative scale $\sim g \sqrt{TM}$. We conjecture that asimilar result should hold if the torus is replaced by any other compactsurface of genus one.

Linde problem in Yang-Mills theory compactified on $\mathbb{R}^2 \times \mathbb{T}^2$