The purpose of this paper is to discuss Lie color triple systems. The cohomology theory of Lie color triple systems is established, then 1-parameter formal deformations and abelian extensions of Lie color triple systems are studied using cohomology.

The theory of cohomologies on multiplicative Hom- $\delta$-Jordan Lie color triple systems is given. As an application, deformations and extensions on the multiplicative Hom- $\delta$-Jordan Lie color triple system are characterized in view of relevant cohomology.