Combining strong coupling and hopping expansion one can derive a dimensionally reduced effective theory of lattice QCD. This theory has a reduced sign problem, is amenable to analytic evaluation and was successfully used to study the cold and dense regime of QCD for sufficiently heavy quarks. We show results from the evaluation of the effective theory for arbitrary $N_c$ up to $\kappa^4$. The inclusion of gauge corrections is also investigated. We find that the onset transition to finite baryon number density steepens with growing $N_c$ even for $T \neq 0$. This suggests that in the large $N_c$ limit the onset transition is first order up to the deconfinement transition. Beyond the onset, the pressure is shown to scale as $p \sim N_c$ through three orders in the hopping expansion, which is characteristic for a phase termed quarkyonic matter in the literature.