  In this paper, we consider compatible Hom-Lie algebras as a twisted version
of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as
Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also
define a cohomology theory for compatible Hom-Lie algebras generalizing the
recent work of Liu, Sheng and Bai. As applications of cohomology, we study
abelian extensions and deformations of compatible Hom-Lie algebras.
