We use the probability distribution function (PDF) of the Lyman α forest flux at z= 2–3, measured from high-resolution UVES/VLT data, and hydrodynamical simulations to obtain constraints on cosmological parameters and the thermal state of the intergalactic medium (IGM) at z∼ 2–3. The observed flux PDF at z= 3 alone results in constraints on cosmological parameters in good agreement with those obtained from the Wilkinson Microwave Anisotropy Probe (WMAP) data, albeit with about a factor of 2 larger errors. The observed flux PDF is best fit with simulations with a matter fluctuation amplitude of σ8= 0.8–0.85 ± 0.07 and an inverted IGM temperature–density relation (γ∼ 0.5–0.75), consistent with our previous results obtained using a simpler analysis. These results appear to be robust to uncertainties in the quasar (quasi-stellar object) continuum placement. We further discuss constraints obtained by a combined analysis of the high-resolution flux PDF and the power spectrum measured from the Sloan Digital Sky Survey (SDSS) Lyman α forest data. The joint analysis confirms the suggestion of an inverted temperature–density relation, but prefers somewhat higher values (σ8∼ 0.9) of the matter fluctuation amplitude than the WMAP data and the best fit to the flux PDF alone. The joint analysis of the flux PDF and power spectrum (as well as an analysis of the power spectrum data alone) prefers rather large values for the temperature of the IGM, perhaps suggesting that we have identified a not yet accounted for systematic error in the SDSS flux power spectrum data or that the standard model describing the thermal state of the IGM at z∼ 2–3 is incomplete.