Equivariant formal deformations of Hom-pre-Lie algebras

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one considering the structure map $\alpha$. We show this new cohomology is deformation cohomology of the Hom-pre-Lie algebra. We also develop cohomology and the associated deformation theory for Hom-pre-Lie algebras in the equivariant context.