  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.
