  We study the ROSAT surface brightness profiles of a sample of 25 distant
(0.3<z<0.83) hot (kT>3.5keV) clusters. For both open and flat cosmological
models, the derived emission measure profiles are scaled according to the
self-similar model of cluster formation. We use the standard scaling relations
of cluster properties with z and temperature, with the empirical slope of the
Mgas-T relation derived by Neumann & Arnaud (2001). Using a \chi^2 test, we
perform a quantitative comparison of the scaled emission measure profiles of
distant clusters with a reference profile derived from the sample of 15 hot
nearby clusters compiled by Neumann & Arnaud (1999), which were found to obey
self-similarity. For a low density flat universe, the distant cluster scaled
data were found to be consistent, both in shape and normalisation, with the
reference local profile.It indicates that hot clusters constitute a homologous
family up to high z, and supports the standard picture of structure formation
for the dark matter component. Because of the intrinsic regularity in the hot
cluster population, the scaled profiles can be used as distance indicators, the
correct cosmology being the one for which the profiles at different redshifts
coincide. The present data allow us to put a tight constraint on Omega0 for a
flat Universe: Omega0=0.40^{+0.15}_{-0.12} at 90% confidence level (statistical
errors only). The critical model was excluded at the 98% confidence level.
Consistently, the observed evolution of the normalisation of the Lx-T relation
was found to comply with the self-similar model for Omega0=0.4, Lambda=0.6. The
constraint derived on Omega0 is in remarkable agreement with the constraint
obtained from SNI or from combined analysis of the power spectrum of the 2dFGRS
and the CMB anisotropy. ABRIDGED
