Abstract We present two measurements of the temperature–density relationship (TDR) of the intergalactic medium (IGM) in the redshift range 2.55 < z < 2.95 using a sample of 13 high-quality quasar spectra and high resolution numerical simulations of the IGM. Our approach is based on fitting the neutral hydrogen column density $N_{\rm H\, \small {I}}$ and the Doppler parameter b of the absorption lines in the Lyα forest. The first measurement is obtained using a novel Bayesian scheme that takes into account the statistical correlations between the parameters characterizing the lower cut-off of the $b\hbox{--}N_{\rm H\, \small {I}}$ distribution and the power-law parameters T0 and γ describing the TDR. This approach yields T0/103 K = 15.6 ± 4.4 and γ = 1.45 ± 0.17 independent of the assumed pressure smoothing of the small-scale density field. In order to explore the information contained in the overall $b\hbox{--}N_{\rm H\, \small {I}}$ distribution rather than only the lower cut-off, we obtain a second measurement based on a similar Bayesian analysis of the median Doppler parameter for separate column-density ranges of the absorbers. In this case, we obtain T0/103 K = 14.6 ± 3.7 and γ = 1.37 ± 0.17 in good agreement with the first measurement. Our Bayesian analysis reveals strong anticorrelations between the inferred T0 and γ for both methods as well as an anticorrelation of the inferred T0 and the pressure smoothing length for the second method, suggesting that the measurement accuracy can in the latter case be substantially increased if independent constraints on the smoothing are obtained. Our results are in good agreement with other recent measurements of the thermal state of the IGM probing similar (over-)density ranges.