  In this paper, we introduce the notions of $f$-frequent hypercyclicity and
${\mathcal F}$-hypercyclicity for $C$-distribution semigroups in separable
Fr\'echet spaces. We particularly analyze the classes of $q$-frequently
hypercyclic $C$-distribution semigroups ($q\geq 1$) and frequently hypercyclic
$C$-distribution semigroups, providing a great number of illustrative examples.
