Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves $\mathcal{F}:a{X}^n+b{Y}^n=Z^n$ 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.