  Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$,
$d\mu=e^{-\phi(x)}d\nu$ be the weighted measure and $\Delta_{p,\phi}$ be the
weighted $p$-Laplacian. In this article we will investigate monotonicity for
the first eigenvalue problem of the weighted $p$-Laplace operator acting on the
space of functions along the Ricci-Bourguignon flow on closed Riemannian
manifolds. We find the first variation formula for the eigenvalues of the
weighted $p$-Laplacian on a closed Riemannian manifold evolving by the
Ricci-Bourguignon flow and we obtain various monotonic quantities. At the end
we find some applications in $2$-dimensional and $3$-dimensional manifolds and
give an example.
