  We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie
algebroids, and systematically describe a cohomology complex by considering
coefficient modules. We define the notion of extensions for hom-Lie-Rinehart
algebras. In the sequel, we deduce a characterisation of low dimensional
cohomology spaces in terms of the group of automorphisms of certain abelian
extension and the equivalence classes of those abelian extensions in the
category of hom-Lie-Rinehart algebras, respectively. We also construct a
canonical example of hom-Lie-Rinehart algebra associated to a given Poisson
algebra and an automorphism.
