Context. Observational studies carried out to calibrate the masses of galaxy clusters often use mass–richness relations to interpret galaxy number counts. Aims. Here, we aim to study the impact of the richness–mass relation modelled with cosmological parameters on mock mass calibrations. Methods. We build a Gaussian process regression emulator of high-mass satellite abundance normalisation and log-slope based on cosmological parameters Ω_m, Ω_b, σ₈, h₀, and redshift z. We train our emulator using Magneticum hydrodynamic simulations that span different cosmologies for a given set of feedback scheme parameters. Results. We find that the normalisation depends, albeit weakly, on cosmological parameters, especially on Ω_m and Ω_b, and that their inclusion in mock observations increases the constraining power of these latter by 10%. On the other hand, the log-slope is ≈1 in every setup, and the emulator does not predict it with significant accuracy. We also show that satellite abundance cosmology dependency differs between full-physics simulations, dark-matter only, and non-radiative simulations. Conclusions. Mass-calibration studies would benefit from modelling of the mass–richness relations with cosmological parameters, especially if the satellite abundance cosmology dependency.