We address the SU(N) Fermi-Hubbard model on a chain, with N the number of degenerate orbitals, or colors, for each fermion. In the limit of both large number of colors N and particles, and small number of sites L ≥2, the model is proved to undergo, for attractive local interaction amplitude U, a ZL symmetry breaking which is reminiscent of the phase transitions in the L- states Potts models. Using a combination of Exact Diagonalization with full SU(N) symmetry, generalized L-levels Holstein-Primakoff transformation, Hartree-Fock method and large-N saddle point approximation of the partition function, we extend the results obtained in [PRA 111, L020201 (2025)] to L ≥3 and finite temperature T > 0. In particular, we show that at T = 0 for U < Uc ∼−1/N, the ground state is L-fold degenerate, while for positive temperatures, the critical temperature is both proportional to N and U, i.e. Tc ∝ −UN, making this phase transition particularly suitable for large-N fermions.