  The observed evolution of the galaxy cluster X-ray integral temperature
distribution function between $z=0.05$ and $z=0.32$ is used in an attempt to
constrain the value of the density parameter, $\Omega_{0}$, for both open and
spatially-flat universes. We estimate the overall uncertainty in the
determination of both the observed and the predicted galaxy cluster X-ray
integral temperature distribution functions at $z=0.32$ by carrying out Monte
Carlo simulations, where we take into careful consideration all the most
important sources of possible error. We include the effect of the formation
epoch on the relation between virial mass and X-ray temperature, improving on
the assumption that clusters form at the observed redshift which leads to an
{\em overestimate} of $\Omega_0$. We conclude that at present both the
observational data and the theoretical modelling carry sufficiently large
associated uncertainties to prevent an unambiguous determination of
$\Omega_{0}$. We find that values of $\Omega_{0}$ around 0.75 are most
favoured, with $\Omega_{0}<0.3$ excluded with at least 90 per cent confidence.
In particular, the $\Omega_{0}=1$ hypothesis is found to be still viable as far
as this dataset is concerned. As a by-product, we also use the revised data on
the abundance of galaxy clusters at $z=0.05$ to update the constraint on
$\sigma_8$ given by Viana & Liddle 1996, finding slightly lower values than
before.
