  We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically
describe a cohomology complex by considering coefficient modules. Furthermore,
we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the
first cohomology space in terms of the group of automorphisms of an $A$-split
abelian extension and the equivalence classes of $A$-split abelian extensions.
Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.
