  The aim of this paper is to study the cohomology of Hom-Leibniz
superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra
of Hom-type and provide as application the computations of the derivations and
second cohomology group. Moreover, we extend to graded case the Takhtajan's
construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from
cohomology of Hom-Leibniz algebras.
