  The main purpose of this paper is to provide a full cohomology of a
Hom-pre-Lie algebra with coefficients in a given representation. This new type
of cohomology exploit strongly the Hom-type structure and fits perfectly with
simultaneous deformations of the multiplication and the homomorphism defining a
Hom-pre-Lie algebra. Moreover, we show that its second cohomology group
classifies abelian extensions of a Hom-pre-Lie algebra by a representation.
