  We perform the first study of the oscillation frequencies of rapidly rotating
neutron stars in alternative theories of gravity, focusing mainly on the
fundamental $f$-modes. We concentrated on a particular class of alternative
theories - the (massive) scalar-tensor theories. The generalization to rapid
rotation is important because on one hand the rapid rotation can magnify the
deviations from general relativity compared to the static case and on the other
hand some of the most efficient emitters of gravitational radiation, such as
the binary neutron star merger remnants, are supposed to be rotating close to
their Kepler (mass-shedding) limits shortly after their formation. We have
constructed several sequences of models starting from the nonrotating case and
reaching up to the Kepler limit, with different values of the scalar-tensor
theory coupling constant and the scalar field mass. The results show that the
deviations from pure Einstein's theory can be significant especially in the
case of nonzero scalar field mass. An important property of the oscillation
modes of rapidly rotating stars is that they can become secularly unstable due
to the emission of gravitational radiation, that is so-called
Chandrasekhar-Friedman-Schutz instability. Such unstable modes are efficient
emitters of gravitational radiation. Our studies show that the inclusion of
nonzero scalar field would decrease the threshold value of the normalized
angular momentum where this instability stars to operate, but the growth time
of the instability seems to be increased compared to pure general relativity.
