  We forecast the potential of the forthcoming X-ray galaxy-cluster survey with
eROSITA to constrain dark-energy models. We focus on spatially-flat
cosmological scenarios with either constant or time-dependent dark-energy
equation-of-state parameters. Fisher information is extracted from the number
density and spatial clustering of a photon-count-limited sample of clusters of
galaxies up to z~2. We consider different scenarios for the availability of (i)
X-ray follow-up observations, (ii) photometric and spectroscopic redshifts, and
(iii) accurate knowledge of the observable -- mass relation down to the scale
of galaxy groups. With about 125,000 clusters (detected with more than 50
photons and with mass M500c > $10^{13} h^{-1}$ Msun) from an average all-sky
exposure of 1.6 ks, eROSITA will give marginalized, one-dimensional, 1-$\sigma$
errors of $\Delta \sigma_8 = \pm~0.008$ ($\sim$1 per cent), $\Delta \Omega_{\rm
M} = \pm~0.006$ (2.2 per cent), $\Delta w_0 = \pm~0.07$ (7 per cent), and
$\Delta w_a = \pm~0.25$ (optimistic scenario) in combination with (and largely
improving upon) current constraints from various cosmological probes (cosmic
microwave background, BAOs, Type Ia SNe). Our findings correspond to a
dark-energy figure of merit in the range of $116-162$ (after the four years of
all-sky survey), making eROSITA one of the first Stage IV experiments to come
on line according to the classification of the Dark Energy Task Force. To
secure improved mass calibrations and to include high-redshift clusters (z >
0.5) as well as objects at the group-mass scale (M500c < 5 $\times~ 10^{13}
h^{-1}$ Msun) will be vital to reach such accuracies.
