  The glitch activity of young pulsars arises from the exchange of angular
momentum between the crust and the interior of the star. Recently, it was
inferred that the moment of inertia of the crust of a neutron star is not
sufficient to explain the observed glitches. Such estimates are presumed in the
Einstein's general relativity in describing the hydrostatic equilibrium of
neutron stars. The crust of the neutron star has a space-time curvature of 14
orders of magnitude larger than that probed in solar system tests. This makes
gravity the weakest constrained physics input in the crust related processes.
We calculate the ratio of the crustal to the total moment of inertia of neutron
stars in the scalar-tensor theory of gravity and the non-perturbative $f({\cal
R})={\cal R}+ a {\cal R}^2$ gravity. We find for the former that the crust to
core ratio of the moment of inertia does not change significantly from what is
inferred in general relativity. For the latter we find that the ratio increases
significantly from what is inferred in general relativity in the case of high
mass objects. Our results suggest that the glitch activity of pulsars may be
used to probe gravity models, although the gravity models explored in this work
are not appropriate candidates.
