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The interest in the thermodynamics of supersymmetric Yang-Mills started afterMaldacena proposed the duality between string theory on AdS backgrounds and thelarge-N limit of SYM theories. One of the motivations to study the thermalproperties of ${\cal N}=4$ supersymmetric Yang-Mills in four dimensions(SYM$_{4,4}$) is that at high temperatures, the weak-coupling limit of thistheory has many similarities with high temperature quantum chromodynamics(QCD). In this proceedings contribution, we review recent calculations of theresummed perturbative free energy of ${\cal N}=4$ supersymmetric Yang-Mills infour spacetime dimensions through second order in the 't Hooft coupling$\lambda$ at finite temperature and zero chemical potential. We compare ourfinal result with prior results obtained in the weak and strong-coupling limitsand construct a generalized Pad\'{e} approximant that interpolates between theweak-coupling result and the large-$N_c$ strong-coupling result.

Some recent advances in the understanding of ${\cal N}=4$ supersymmetric Yang-Mills thermodynamics