  In this paper, we introduce twisted relative Rota-Baxter operators on a
Leibniz algebra as a generalization of twisted Poisson structures. We define
the cohomology of a twisted relative Rota-Baxter operator $K$ as the
Loday-Pirashvili cohomology of a certain Leibniz algebra induced by $K$ with
coefficients in a suitable representation. Then we consider formal deformations
of twisted relative Rota-Baxter operators from cohomological points of view.
Finally, we introduce and study NS-Leibniz algebras as the underlying structure
of twisted relative Rota-Baxter operators.
