  Weighted Rota-Baxter operators on associative algebras are closely related to
modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal
bialgebras, and play an important role in mathematical physics. For any
$\lambda \in {\bf k}$, we construct a differential graded Lie algebra whose
Maurer-Cartan elements are given by $\lambda$-weighted relative Rota-Baxter
operators. Using such characterization, we define the cohomology of a
$\lambda$-weighted relative Rota-Baxter operator $T$, and interpret this as the
Hochschild cohomology of a suitable algebra with coefficients in an appropriate
bimodule. We study linear, formal and finite order deformations of $T$ from
cohomological points of view. Among others, we introduce Nijenhuis elements
that generate trivial linear deformations and define a second cohomology class
to any finite order deformation which is the obstruction to extend the
deformation. In the end, we also consider the cohomology of $\lambda$-weighted
relative Rota-Baxter operators in the Lie case and find a connection with the
case of associative algebras.
