  Context: Modelling satellite galaxy abundance $N_s$ in Galaxy Clusters (GCs)
is a key element in modelling the Halo Occupation Distribution (HOD), which
itself is a powerful tool to connect observational studies with numerical
simulations. Aims: To study the impact of cosmological parameters on satellite
abundance both in cosmological simulations and in mock observations. Methods:
We build an emulator (HODEmu, \url{https://github.com/aragagnin/HODEmu/}) of
satellite abundance based on cosmological parameters $\Omega_m, \Omega_b,
\sigma_8, h_0$ and redshift $z.$ We train our emulator using \magneticum
hydrodynamic simulations that span 15 different cosmologies, each over $4$
redshift slices between $0<z<0.5,$ and for each setup we fit normalisation $A$,
log-slope $\beta$ and Gaussian fractional-scatter $\sigma$ of the $N_s-M$
relation. The emulator is based on multi-variate output Gaussian Process
Regression (GPR). Results: We find that $A$ and $\beta$ depend on cosmological
parameters, even if weakly, especially on $\Omega_m,$ $\Omega_b.$ This
dependency can explain some discrepancies found in literature between satellite
HOD of different cosmological simulations (Magneticum, Illustris, BAHAMAS). We
also show that satellite abundance cosmology dependency differs between
full-physics (FP) simulations, dark-matter only (DMO), and non-radiative
simulations. Conclusions: This work provides a preliminary calibration of the
cosmological dependency of the satellite abundance of high mass halos, and we
showed that modelling HOD with cosmological parameters is necessary to
interpret satellite abundance, and we showed the importance of using FP
simulations in modelling this dependency.
