In this paper, first we introduce the notion of crossed homomorphisms on Lie conformal algebras, then we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Lie conformal algebras. This permits us to define cohomology for crossed homomorphisms. As an application, we study infinitesimal deformations, formal deformations and extendibility of finite order deformations of crossed homomorphism between Lie conformal algebras in terms of the cohomology theory.
Basdouri et al. (Wed,) studied this question.