  An FI-module $V$ over a commutative ring $\bf{k}$ encodes a sequence
$(V_n)_{n \geq 0}$ of representations of the symmetric groups
$(\mathfrak{S}_n)_{n \geq 0}$ over $\bf{k}$. In this paper, we show that for a
"finitely generated" FI-module $V$ over a field of characteristic $p$, the
cohomology groups $H^t(\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We
describe a recursive way to calculate the period and the periodicity range and
show that the period is always a power of $p$. As an application, we show that
if $\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\geq
2$ and $\mathit{conf}_n(\mathcal{M})$ is the configuration space of unordered
$n$-tuples of distinct points in $\mathcal{M}$ then the mod-$p$ cohomology
groups $H^{t}(\mathit{conf}_n(\mathcal{M}),\bf{k})$ are eventually periodic in
$n$ with period a power of $p$.
