  The gravitational aether theory is a modification of general relativity that
decouples vacuum energy from gravity, and thus can potentially address the
cosmological constant problem. The classical theory is distinguishable from
general relativity only in the presence of relativistic pressure (or
vorticity). Since the interior of neutron stars has high pressure and as their
mass and radius can be measured observationally, they are the perfect
laboratory for testing the validity of the aether theory. In this paper, we
solve the equations of stellar structure for the gravitational aether theory
and find the predicted mass-radius relation of non-rotating neutron stars using
two different realistic proposals for the equation of state of nuclear matter.
We find that the maximum neutron star mass predicted by the aether theory is
12% - 16% less than the maximum mass predicted by general relativity assuming
these two equations of state. We also show that the effect of aether is similar
to modifying the equation of state in general relativity. The effective
pressure of the neutron star given by the aether theory at a fiducial density
differs from the values given by the two nuclear equations of state to an
extent that can be constrained using future gravitational wave observations of
neutron stars in compact systems. This is a promising way to test the aether
theory if further progress is made in constraining the equation of state of
nuclear matter in densities above the nuclear saturation density.
