  We re-examine the constraints which can be robustly obtained from the
observed temperature function of X-ray cluster of galaxies. The cluster mass
function has been thoroughly studied in simulations and analytically, but a
direct simulation of the temperature function is presented here for the first
time. Adaptive hydrodynamic simulations using the cosmological Moving Mesh
Hydro code of Pen (1997a) are used to calibrate the temperature function for
different popular cosmologies. Applying the new normalizations to the
present-day cluster abundances, we find $\sigma_8=0.53\pm 0.05
\Omega_0^{-0.45}$ for a hyperbolic universe, and $\sigma_8=0.53\pm 0.05
\Omega_0^{-0.53}$ for a spatially flat universe with a cosmological constant.
The simulations followed the gravitational shock heating of the gas and dark
matter, and used a crude model for potential energy injection by supernova
heating. The error bars are dominated by uncertainties in the heating/cooling
models. We present fitting formulae for the mass-temperature conversions and
cluster abundances based on these simulations.
