  Bihom-associative algebras have been recently introduced in the study of
group hom-categories. In this paper, we introduce a Hochschild type cohomology
for bihom-associative algebras with suitable coefficients. The underlying
cochain complex (with coefficients in itself) can be given the structure of an
operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber
structure. We show that this cohomology also control corresponding formal
deformations. Finally, we introduce bihom-associative algebras up to homotopy
and show that some particular classes of these homotopy algebras are related to
the above Hochschild cohomology.
