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

Cohomology of Hom-Leibniz and $n$-ary Hom-Nambu-Lie superalgebras