We use the ROSAT Deep Cluster Survey (RDCS) to trace the evolution of the cluster abundance out to $zsimeq 0.8$ and constrain cosmological models. We resort to a phenomenological prescription to convert masses into $X$-ray fluxes and apply a maximum-likelihood approach to the RDCS redshift- and luminosity-distribution. We find that, even changing the shape and the evolution on the $L_{bol}$-$T_X$ relation within the observational uncertainties, a critical density Universe is always excluded at more than $3sigma$ level. By assuming a non-evolving $X$-ray luminosity-temperature relation with shape $L_{bol}propto T_X^3$, it is $Omega_m=0.35^{+0.35}_{-0.25}$ and $sigma_8=0.76^{+0.38}_{-0.14}$ for flat models, with uncertainties corresponding to $3sigma$ confidence levels.
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