Non-gravitational processes, such as feedback from galaxies and their active nuclei, are believed to have injected excess entropy into the intracluster gas, and therefore to have modified the density profiles in galaxy clusters during their formation. Here we study a simple model for this so-called preheating scenario, and ask (i) whether it can simultaneously explain both global X-ray scaling relations and number counts of galaxy clusters, and (ii) whether the amount of entropy required evolves with redshift. We adopt a baseline entropy profile that fits recent hydrodynamic simulations, modify the hydrostatic equilibrium condition for the gas by including approx. 20% non-thermal pressure support, and add an entropy floor K_0 that is allowed to vary with redshift. We find that the observed luminosity-temperature (L-T) relations of low-redshift (z=0.05) HIFLUGCS clusters and high-redshift (z=0.8) WARPS clusters are best simultaneously reproduced with an evolving entropy floor of K_0(z)=341(1+z)^{-0.83}h^{-1/3} keV cm^2. If we restrict our analysis to the subset of bright (kT > 3 keV) clusters, we find that the evolving entropy floor can mimic a self-similar evolution in the L-T scaling relation. This degeneracy with self-similar evolution is, however, broken when (0.5 < kT < 3 keV) clusters are also included. The approx. 60% entropy increase we find from z=0.8 to z=0.05 is roughly consistent with that expected if the heating is provided by the evolving global quasar population. Using the cosmological parameters from the WMAP 3-year data with sigma_8=0.76, our best-fit model underpredicts the number counts of the X-ray galaxy clusters compared to those derived from the 158 deg^2 ROSAT PSPC survey. Treating sigma_8 as a free parameter, we find a best-fit value of sigma_8=0.80+/- 0.02.
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