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For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, weestablish necessary and sufficient conditions for $\mathcal{F}$ to be$\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system ofplane cubics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determineexplicit formulas for the number $N_q(\mathcal{F})$ of $\mathbb{F}_q$-rationalpoints 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.

Frobenius nonclassicality of Fermat curves with respect to cubics