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At finite temperature the free energy density of ${\cal N}=4$ supersymmetricYang-Mills can be calculated using resummed perturbation theory through theorder $\lambda^{5/2}$. Effective field theory methods provide a usefulalternative approach to streamline these calculations. In this proceedingscontribution, I review recent work with my collaborators where we usedeffective field theory methods to calculate the free energy density of ${\calN}=4$ supersymmetric Yang-Mills in four spacetime dimensions through secondorder in the 't Hooft coupling $\lambda$. At this order the contributions tothe free energy density come from the hard scale $T$ and the soft scale$\sqrt{\lambda}T$. The contribution from the scale $T$ enters through thecoefficients in the effective Lagrangian obtained by dimensional reduction andthe effects of the scale $gT$ can be calculated using perturbative methods inthe effective theory.

Effective field theory treatment of ${\cal N}=4$ supersymmetric Yang-Mills thermodynamics