  We construct rapidly rotating neutron star models in scalar-tensor theories
with a massive scalar field. The fact that the scalar field has nonzero mass
leads to very interesting results since the allowed range of values of the
coupling parameters is significantly broadened. These deviations from pure
general relativity can be very large for values of the parameters that are in
agreement with the observations. The rapid rotation can magnify the differences
several times compared to the static case. The universal relations between the
normalized moment of inertia and quadrupole moment are also investigated both
for the slowly and rapidly rotating cases. The results show that these
relations are still EOS independent up to a large extend and the deviations
from pure general relativity can be large. This places the massive
scalar-tensor theories amongst the few alternative theories of gravity that can
be tested via the universal $I$-Love-$Q$ relations.
