  In this paper, we firstly construct an $L_\infty[1]$-algebra via the method
of higher derived brackets, whose Maurer-Cartan elements correspond to relative
$\Omega$-family Rota-Baxter algebras structures of weight $\lambda$. For a
relative $\Omega$-family Rota-Baxter algebra of weight $\lambda$, the
corresponding twisted $ L_{\infty}[1] $-algebra controls its deformations,
which leads to the cohomology theory of relative $\Omega$-family Rota-Baxter
algebras of weight $\lambda$. Moreover, we also obtain the corresponding
results for absolute $\Omega$-family Rota-Baxter algebras of weight $\lambda$
from the relative version. At last, we study formal deformations of relative
(resp. absolute) $\Omega$-family Rota-Baxter algebras of weight $\lambda$,
which can be explained by the lower degree cohomology groups.
