  We compare the improved measurement of the Lya forest flux probability
distribution at 1.7<z<3.2 presented by Kim et al. (2007) to a large set of
hydrodynamical simulations of the Lya forest with different cosmological
parameters and thermal histories. The simulations are in good agreement with
the observational data if the temperature-density relation for the low density
intergalactic medium (IGM), T=T_0 Delta^{gamma-1}, is either close to
isothermal or inverted (gamma<1). Our results suggest that the voids in the IGM
may be significantly hotter and the thermal state of the low density IGM may be
substantially more complex than is usually assumed at these redshifts. We
discuss radiative transfer effects which alter the spectral shape of ionising
radiation during the epoch of HeII reionisation as a possible physical
mechanism for achieving an inverted temperature-density relation at z~3.
