A model of intergalactic medium heated by QSOs and cooled by the expansion of the universe and Compton cooling is studied in the framework of a Friedmann-Robertson-Walker universe. Cosmological evolution functions of the comoving density of QSO's as well as the case of no evolution are considered. The theoretical X-ray background spectrum (through thermal bremsstrahlung) and Compton y parameter are calculated including relativistic corrections in the electron-electron, electron-proton and electron-photon interactions. The observed X-ray background and the upperlimit of the Compton parameter y(COBE) given by the COBE satellite are used to adjust, for each value of reheating redshifts z(c) ranging from 0.1 to 5.0, the present values of the temperature T0 and density no of the intergalactic gas. For z(c) > 0.25, when the theoretical X-ray spectrum fits the observed one, the adjusted values of T0 and no imply in y > y(COBE). On the other hand, when T0 and n0 are consistent with y(COBE), the calculated X-ray spectrum is lower than the observed one. Unless 100% of the observed X-ray background is due to discrete sources and if the intergalactic medium contributes more than 2.5% to such,background we come to the interesting result that the medium must have been heated at z(c) < 1. In this case we shall have to explain the high energy rates necessary to heat the intergalactic medium. For z(c) less than or similar 0.25, it is possible to find values of T0 and n0 such that both the calculated X-ray background and the y parameter simultaneously reproduce the corresponding observed values. However, in this case, unless it could be shown to be otherwise by future observations or theoretical studies, it seems that the model of hot intergalactic medium is not plausible because of the high energies required to heat the intergalactic gas.