  In this paper, first, we introduce a notion of modified Rota-Baxter Lie
algebras of weight $\mathrm{\lambda}$ with derivations (or simply modified
Rota-Baxter LieDer pairs) and their representations. Moreover, we investigate
cohomologies of a modified Rota-Baxter LieDer pairs with coefficients in a
suitable representation. As applications, we study formal one-parameter
deformations and abelian extensions of modified Rota-Baxter LieDer pairs.
