  We use the X-ray power spectrum of the ROSAT all-sky survey in the R6 band
(approximately 0.9-1.3 keV) to set an upper limit on the galaxy cluster power
spectrum. The cluster power spectrum is modelled with a minimum number of
robust assumptions regarding the structure of the clusters. The power spectrum
of ROSAT sets an upper limit on the Omega_m-sigma_8 plane which excludes all
the models with sigma_8 above sigma_8 = 0.5/(Omega_m^0.38) in a flat LCDM
universe. We discuss the possible sources of systematic errors in our
conclusions, mainly dominated by the assumed L_x-T relation. Alternatively,
this relation could be constrained by using the X-ray power spectrum, if the
cosmological model is known. Our conclusions suggest that only models with a
low value of sigma_8 (sigma_8 < 0.8 for Omega_m = 0.3) may be compatible with
our upper limit. We also find that models predicting lower luminosities in
galaxy clusters are favoured. Reconciling our cosmological constraints with
these arising by other methods might require either a high entropy floor or
wide-spread presence of cooling flows in the low-redshift clusters.
