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We propose a general formula for higher order corrections to the value of theHagedorn temperature of a class of holographic confining gauge theories in thestrong coupling expansion. Inspired by recent proposals in the literature, theformula combines the sigma-model string expansion with an effective approach.In particular, it includes the sigma-model contributions to the Hagedorntemperature at next-to-next-to leading order, which are computed in fullgenerality. For ${\cal N}=4$ SYM on $S^3$ our result agrees with numericalfield theory estimates with excellent precision. We use the general formula topredict the value of the Hagedorn temperature for ABJM on $S^2$ and for thedual of purely RR global $AdS_3$.

Higher Order Corrections to the Hagedorn Temperature at Strong Coupling