  In this paper, we have extended the entropy-driven model of cluster evolution
developed by Bower (1997) in order to be able to predict the evolution of
galaxy clusters for a range of cosmological scenarios. We have applied this
model to recent measurements of the evolution of the L_x-T normalisation and
X-ray luminosity function in order to place constraints on cosmological
parameters. We find that these measurements alone do not select a particular
cosmological frame-work. An additional constraint is required on the effective
slope of the power spectrum to break the degeneracy that exists between this
and the background cosmology. We have therefore included a theoretical
calculation of the \Omega_0 dependence on the power spectrum, based on the cold
dark matter paradigm, which infers \Omega_0<0.55 (0.1<\Omega_0<0.7 for
\Omega_0+\Lambda_0=1), at the 95% confidence level. Alternatively, an
independent measurement of the slope of the power spectrum from galaxy
clustering requires \Omega_0<0.6 (\Omega_0<0.65 for \Omega_0+\Lambda_0=1),
again to 95% confidence. The rate of entropy evolution is insensitive to the
values of \Omega_0 considered, although is sensitive to changes in the
distribution of the intracluster medium.
