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In this paper, we firstly construct an $L_\infty[1]$-algebra via the methodof higher derived brackets, whose Maurer-Cartan elements correspond to relative$\Omega$-family Rota-Baxter algebras structures of weight $\lambda$. For arelative $\Omega$-family Rota-Baxter algebra of weight $\lambda$, thecorresponding twisted $ L_{\infty}[1] $-algebra controls its deformations,which leads to the cohomology theory of relative $\Omega$-family Rota-Baxteralgebras of weight $\lambda$. Moreover, we also obtain the correspondingresults 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.

Deformations and cohomology theory of $\Omega$-family Rota-Baxter algebras of arbitrary weight