Aims. Recent analyses of the intergalactic UV background by means of the Hell Lya forest assume that Hell and H absorption features have the same line widths. We omit this assumption to investigate possible effects of thermal line broadening on the inferred He II/HIratio η and to explore the potential of intergalactic Hell observations to constrain the thermal state of the intergalactic medium. Methods. Deriving a simple relation between the column density and the temperature of an absorber based on the temperature-density relation T T o (1+ δ) γ-1 we develop a procedure to fit To, y, and η simultaneously by modeling the observed spectra with Doppler profiles. In an alternative approach the temperature T of an absorber, the He II/HIratio η, and the redshift scale of η variations are estimated simultaneously by optimizing the Doppler parameters of He II Results. Testing our procedure with artificial data shows that well-constrained results can be obtained only if the signal-to-noise ratio in the Hell forest is S/N ≥ 20. Additionally, ambiguities in the line profile decomposition may result in significant systematic errors. Thus, it is impossible to give an estimate of the temperature-density relation with the He IIdata available at present (S/IN ∼ 5). However, we find that only 45% of the lines in our sample favor turbulent line widths while the remaining lines are probably affected by thermal broadening. Furthermore, the inferred η values are on average about 0.05 dex larger if a thermal component is taken into account, and their distribution is 46% narrower in comparison to a purely turbulent fit. Therefore, variations of η on a 10% level may be related to the presence of thermal line broadening. The apparent correlation between the strength of the H I absorption and the ηvalue, which has been found in former studies assuming turbulent line broadening, essentially disappears if thermal broadening is taken into account. In the redshift range 2.58 < z < 2.74 towards the quasars HE 2347-4342 and HS 1700+6416 we obtain η ≈100 and slightly larger. In the same redshift range the far-UV spectrum ofHS 1700+6416 is best and we estimate a mean value of log η2.11 ±0.32 taking into account combined thermal and turbulent broadening.