  Smith and Van den Bergh introduced the notion of finite F-representation type
as a characteristic $p$ analogue of the notion of finite representation type.
In this paper, we prove two finiteness properties of rings with finite
F-representation type. The first property states that if $R=\bigoplus_{n \ge
0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type,
then for every non-zerodivisor $x \in R$, $R_x$ is generated by $1/x$ as a
$D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with
finite F-representation type, then $H_I^n(R)$ has only finitely many associated
primes for any ideal $I$ of $R$ and any integer $n$. We also include a result
on the discreteness of F-jumping exponents of ideals of rings with finite
(graded) F-representation type as an appendix.
