  We describe how to calculate the Hagedorn temperature of $\mathcal{N}=4$ SYM
theory and type IIB superstring theory on $AdS_5\times S^5$ via the Quantum
Spectral Curve (QSC) -- providing further details on our previous letters
1706.03074 and 1803.04416. We solve the QSC equations perturbatively at weak 't
Hooft coupling $\lambda$ up to seven-loop order and numerically at finite
coupling, finding that the perturbative results can be expressed in terms of
single-valued harmonic polylogarithms. Moreover, we generalize the QSC to
describe 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) agrees
with the one in $\mathcal{N}=4$ SYM theory at any value of $\lambda$.
