Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as examples... In this article, we introduce a notion of Frobenius $n$-exangulated categories which are a generalization of Frobenius $n$-exact categories. We show that the stable category of a Frobenius $n$-exangulated category is an $(n+2)$-angulated category. As an application, this result generalizes the work by Jasso. We provide a class of $n$-exangulated categories which are neither $n$-exact categories nor $(n+2)$-angulated categories. Finally, we discuss an application of the main results and give some examples illustrating it. (read more)