  We analyze $f$-frequently hypercyclic, $q$-frequently hypercyclic ($q> 1$)
and frequently hypercyclic $C_{0}$-semigroups ($q=1$) defined on complex
sectors, working in the setting of separable infinite-dimensional Fr\'echet
spaces. Some structural results of ours are given for a general class of
${\mathcal F}$-frequently hypercyclic $C_{0}$-semigroups, as well. We
investigate generalized frequently hypercyclic translation semigroups and
generalized frequently hypercyclic semigroups induced by semiflows on weighted
function spaces. Several illustrative examples are presented.
