  In this paper, we introduce the representation of anti-pre-Lie algebras and
give the second cohomology group of anti-pre-Lie algebras. As applications,
first, we study linear deformations of anti-pre-Lie algebras. The notion of a
Nijenhuis operator on an anti-pre-Lie algebra is introduced which can generate
a trivial linear deformation of an anti-pre-Lie algebra. Then, we study formal
deformations of anti-pre-Lie algebras. We show that the infinitesimal of a
formal deformation is a 2-cocycle with the coefficients in the regular
representation and depends only on its cohomology class. Moreover, if the
second cohomology group $H^2(A;A)$ is trivial, then the anti-pre-Lie algebra is
rigid. Finally, we introduce the notion of abelian extensions. We show that
abelian extensions are classified by the second cohomology group $H^2(A;V)$.
