  The aim of this paper is to provide a cohomology of $n$-Hom-Lie color
algebras governing one parameter formal deformations. Then, we study formal
deformations of a $n$-Hom-Lie color algebra and introduce the notion of
Nijenhuis operator on an $n$-Hom-Lie color algebra, which could give rise to
infinitesimally trivial $(n-1)$-order deformations. Furthermore, in connection
with Nijenhuis operators we introduce and discuss the notion of a product
structure on $n$-Hom-Lie color algebras.
