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Rota-Baxter operators have been paid much attention in the last few decadesas they have many applications in mathematics and physics. In this paper, ourobject of study is modified Rota-Baxter operators on Leibniz algebras. Weinvestigate modified Rota-Baxter Leibniz algebras from the cohomological pointof view. We study a one-parameter formal deformation theory of modifiedRota-Baxter Leibniz algebras and define the associated deformation cohomologythat controls the deformation. Finally, as an application, we characterizeequivalence classes of abelian extensions in terms of second cohomology groups.

Cohomology of modified Rota-Baxter Leibniz algebra of weight $\lambda$