  Cosmological studies have now entered Stage IV according to the Dark Energy
Task Force prescription, thanks to new missions (Euclid, Rubin Observatory,
SRG/eROSITA) that are expected to provide the required ultimate accuracy in the
dark energy (DE) equation of state (EoS). However, none of these projects have
the power to systematically unveil the galaxy cluster population at $z>1$.
There therefore remains the need for an ATHENA-like mission to run independent
cosmological investigations and scrutinise the consistency between the results
from the $0<z<1$ and $1<z<2$ epochs. We study the constraints on the DE EoS and
on primordial non-Gaussanities for typical X-ray cluster surveys executed by
ATHENA. We consider two survey designs: 50 deg$^2$ at 80ks (survey A) and 200
deg$^2$ at 20ks (survey B). We analytically derive cluster counts in a space of
observable properties, and predict the cosmological potential of the
corresponding samples with a Fisher analysis. The achieved depth allows us to
unveil the halo mass function down to the group scale out to $z=2$. We predict
the detection of thousands of clusters down to a few 10$^{13} h^{-1}
M_{\odot}$, in particular 940 and 1400 clusters for surveys A and B,
respectively, at $z>1$. Such samples will allow a detailed modelling of the
evolution of cluster physics along with a standalone cosmological analysis. Our
results suggest that survey B has the optimal design as it provides greater
statistics. Remarkably, high-$z$ clusters, despite representing 15% or less of
the full samples, allow a significant reduction of the uncertainty on the
cosmological parameters: $\Delta w_a$ is reduced by a factor of 2.3 and $\Delta
f_{NL}^{loc}$ by a factor of 3. Inventorying the high-$z$ X-ray cluster
population can play a crucial role in ensuring overall cosmological
consistency. This will be the major aim of future new-generation ATHENA-like
missions.
