  Given a Lie algebroid with a representation, we construct a graded Lie
algebra whose Maurer-Cartan elements characterize relative Rota-Baxter
operators on Lie algebroids. We give the cohomology of relative Rota-Baxter
operators and study infinitesimal deformations and extendability of order $n$
deformations to order $n+1$ deformations of relative Rota-Baxter operators in
terms of this cohomology theory. We also construct a graded Lie algebra on the
space of multi-derivations of a vector bundle whose Maurer-Cartan elements
characterize left-symmetric algebroids. We show that there is a homomorphism
from the controlling graded Lie algebra of relative Rota-Baxter operators on
Lie algebroids to the controlling graded Lie algebra of left-symmetric
algebroids. Consequently, there is a natural homomorphism from the cohomology
groups of a relative Rota-Baxter operator to the deformation cohomology groups
of the associated left-symmetric algebroid. As applications, we give the
controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg
structures on left-symmetric algebroids.
