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We describe how to calculate the Hagedorn temperature of $\mathcal{N}=4$ SYMtheory and type IIB superstring theory on $AdS_5\times S^5$ via the QuantumSpectral Curve (QSC) -- providing further details on our previous letters1706.03074 and 1803.04416. We solve the QSC equations perturbatively at weak 'tHooft coupling $\lambda$ up to seven-loop order and numerically at finitecoupling, finding that the perturbative results can be expressed in terms ofsingle-valued harmonic polylogarithms. Moreover, we generalize the QSC todescribe the Hagedorn temperature in the presence of chemical potentials.Finally, we show that the Hagedorn temperature in certain deformations of$\mathcal{N}=4$ SYM theory (real-$\beta$ and $\gamma_i$ deformation) agreeswith the one in $\mathcal{N}=4$ SYM theory at any value of $\lambda$.

Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations