  We investigate the perturbative expansion in $SU(3)$ Yang-Mills theory
compactified on $\mathbb{R}^2\times \mathbb{T}^2$ where the compact space is a
torus $\mathbb{T}^2= S^1_{\beta}\times S^1_{L}$, with $S^1_{\beta}$ being a
thermal 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. A
Linde-type analysis indicates that perturbative calculations for the pressure
in this theory break down already at order $\mathcal{O}(g^2)$ due to the
presence of a non-perturbative scale $\sim g \sqrt{TM}$. We conjecture that a
similar result should hold if the torus is replaced by any other compact
surface of genus one.
