  This paper is devoted to studying deformation, cohomology theory of
Rota-Baxter pre-Lie algebras of arbitrary weights. First we give the notion of
a new representation of a Rota-Baxter pre-Lie algebra of arbitrary weight and
define the cohomology theory of a Rota-Baxter pre-Lie algebra of arbitrary
weight. Then we study formal deformations by lower degree cohomology groups.
Finally, we classify abelian extensions of Rota-Baxter pre-Lie algebras of
arbitrary weight using the second cohomology group, and classify skeletal
Rota-Baxter pre-Lie 2-algebra of arbitrary weight using the third cohomology
group as applications.
