  The evolution of the galaxy cluster abundance depends sensitively on the
value of the cosmological density parameter, Omega_0. Recent ASCA data are used
to quantify this evolution as measured by the X-ray temperature function. A
chi^2 minimisation fit to the cumulative temperature function, as well as a
maximum likelihood estimate (which requires additional assumptions about
cluster luminosities), lead to the estimate Omega_0 \approx 0.45+/-0.2 (1-sigma
statistical error). Various systematic uncertainties are considered, none of
which enhance significantly the probability that Omega_0=1. These conclusions
hold for models with or without a cosmological constant. The statistical
uncertainties are at least as large as the individual systematic errors that
have been considered here, suggesting that additional temperature measurements
of distant clusters will allow an improvement in this estimate. An alternative
method that uses the highest redshift clusters to place an upper limit on
Omega_0 is also presented and tentatively applied, with the result that
Omega_0=1 can be ruled out at the 98 per cent confidence level. Whilst this
method does not require a well-defined statistical sample of distant clusters,
there are still modelling uncertainties that preclude a firmer conclusion at
this time.
