  Ho\v{r}ava-Lifshitz gravity is an alternative theory to general relativity
which breaks Lorentz invariance in order to achieve an ultraviolet complete and
power-counting renormalizable theory of gravity. In the low energy limit,
Ho\v{r}ava-Lifshitz gravity coincides with a vector-tensor theory known as
khronometric gravity. The deviation of khronometric gravity from general
relativity can be parametrized by three coupling constants: $\alpha$, $\beta$,
and $\lambda$. Solar system experiments and gravitational wave observations
impose stringent bounds on $\alpha$ and $\beta$, while $\lambda$ is still
relatively unconstrained ($\lambda\lesssim 0.01$). In this paper, we study
whether one can constrain this remaining parameter with neutron star
observations through the universal I-Love-Q relations between the moment of
inertia (I), the tidal Love number (Love), and the quadrupole moment (Q), which
are insensitive to details in the nuclear matter equation of state. To do so,
we perturbatively construct slowly-rotating and weakly tidally-deformed neutron
stars in khronometric gravity. We find that the I-Love-Q relations are
independent of $\lambda$ in the limit $(\alpha,\beta) \to 0$. Although some
components of the field equations depend on $\lambda$, we show through
induction and a post-Minkowskian analysis that slowly-rotating neutron stars do
not depend on $\lambda$ at all. Tidally deformed neutron stars, on the other
hand, are modified in khronometric gravity (though the usual Love number is not
modified, as mentioned earlier), and there are potentially new, non-GR Love
numbers, though their observability is unclear. These findings indicate that it
may be difficult to constrain $\lambda$ with rotating/tidally-deformed neutron
stars.
