  For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, we
establish necessary and sufficient conditions for $\mathcal{F}$ to be
$\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of
plane cubics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine
explicit formulas for the number $N_q(\mathcal{F})$ of $\mathbb{F}_q$-rational
points on $\mathcal{F}$. For the remaining Fermat curves, nice upper bounds for
$N_q(\mathcal{F})$ are immediately given by the St\"ohr-Voloch Theory.
