The high temperature expansion is an analytical tool to study critical phenomena in statistical mechanics. We apply this method to 3d effective theories of Polyakov loops, which have been derived from 4d lattice Yang-Mills by means of resummed strong coupling expansions. In particular, the Polyakov loop susceptibility is computed as a power series in the effective couplings. A Padé analysis then provides the location of the phase transition in the effective theory, which can be mapped back to the parameters of 4d Yang-Mills. Our purely analytical results for the critical couplings $\beta_c(N_\tau)$ agree to better than $10\%$ with those from Monte Carlo simulations. For the case of $SU(2)$, also the critical exponent $\gamma$ is predicted accurately, while a first-order nature as for $SU(3)$ cannot be identified by a Padé analysis. The method can be generalised to include fermions and finite density.