  In this paper, we introduce the representation of modified
$\lambda$-differential $3$-Lie algebras and define the cohomology of modified
$\lambda$-differential $3$-Lie algebras with coefficients in a representation.
As applications of the proposed cohomology theory, we study linear
deformations, abelian extensions and $T^*$-extensions of modified
$\lambda$-differential $3$-Lie algebras.

Rings and Algebras