  I study the fermionic $U(N)$ Gross-Neveu model at imaginary chemical
potential and finite temperature for odd $d$ dimensions, in the strong coupling
regime, by using the gap (saddle point) equation for the fermion condensate of
the model. This equation describes the phase transitions from weak to strong
coupling regime. I point out that the higher odd dimensional gap equations are
linear combinations of the lower dimensional equations in a way that as the
dimension of the model increases the lower dimensions are weaker coupled but
still in the strong coupling regime. Interestingly, at a specific value of the
chemical potential, exactly in the middle of the thermal windows that separate
the fermionic from the bosonic (condensed) state of the fermions, I find the
mass of the fermion condensate for $d=3,5,7,9$. An anomaly occurs at the $5$
dimensional theory where it is stronger coupled against other theories in
higher dimensions and lower energy. The main idea of this work is that the
cut-off $\Lambda$ regulator for the UV divergent parts of the fermion mass
saddle point equation, plays the role of a physical parameter. This idea is
based on the identity of the asymptotic freedom of the Gross-Neveu model as a
toy model for QCD.
