  In this paper we define a new cohomology for multiplicative Hom-associative
algebras, which generalize Hochschild cohomology and fits with deformations of
Hom-associative algebras including the structure map $\alpha$. It is a
generalization of the known Hochschild-type cohomology for Hom-associative
algebras defined by Amar, Ejbehi and Makhlouf. Moreover, we provide various
observations and similarly a new cohomology of Hom-bialgebras extending the
Gerstenhaber-Schack cohomology for Hom-bialgebras given by Dekkar and Makhlouf.
