We describe how to calculate the Hagedorn temperature of N = 4 SYM theory and type IIB superstring theory on AdS(5) x S-5 via the Quantum Spectral Curve (QSC) - providing further details on our previous letters [1] and [2]. 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 N = 4 SYM theory (real-beta and gamma(i) deformation) agrees with the one in N = 4 SYM theory at any value of lambda.