  Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold ($n>1$) whose
metric $g(t)$ evolves by the generalized abstract geometric flow. This paper
discusses the evolution, monotonicity and differentiability for the first
eigenvalue of the $p$-Laplacian on $(M,g(t))$ with respect to time evolution.
We prove that the first nonzero $p$-eigenvalue is monotone nondecreasing along
the flow under certain geometric condition and that the first eigenvalue is
differentiable almost everywhere. When $p=2$, we recover the corresponding
results for the usual Laplace-Beltrami operator. Our results provide a unified
approach to the study of $p$-eigenvalue under various geometric flows
