  In this paper, we first introduce the concept of Rota-Baxter family
BiHom-$\Omega$-associative algebras, then we define the cochain complex of
BiHom-$\Omega$-associative algebras and verify it via Maurer-Cartan methods.
Next, we further introduce and study the cohomology theory of Rota-Baxter
family BiHom-$\Omega$-associative algebras of weight $\lambda$ and show that
this cohomology controls the corresponding deformations. Finally, we study
abelian extensions of Rota-Baxter family BiHom-$\Omega$-associative algebras in
terms of the second cohomology group.
