The amplitude of density perturbations, for the currently-favoured ΛCDM cosmology, is constrained using the observed properties of galaxy clusters. The catalogue used is that of Ikebe et al. The relation of cluster temperature to mass is obtained via N-body/hydrodynamical simulations including radiative cooling and pre-heating of cluster gas, which we have previously shown to reproduce well the observed temperature–mass relation in the innermost parts of clusters. We generate and compare mock catalogues via a Monte Carlo method, which allows us to constrain the relation between X-ray temperature and luminosity, including its scatter, simultaneously with cosmological parameters. We find a luminosity–temperature relation in good agreement with the results of Ikebe et al., while for the matter power spectrum normalization, we find σ8= 0.78+0.30−0.06 at 95 per cent confidence for Ω0= 0.35. Scaling to the Wilkinson Microwave Anisotropy Probe central value of Ω0= 0.27 would give a best-fitting value of σ8≃ 0.9.