  Modified gravity models often contain modes that couple to normal matter and
propagate with slightly less than the speed of light. High-energy cosmic rays
then lose energy due to Cherenkov radiation, which constrains such models. This
is also true for some MOND (Modified Newtonian Dynamics) models. However, these
constraints are difficult to make precise because MOND is inherently non-linear
and because the results may depend on the specific high-acceleration behavior
of these models, i.e. the behavior outside the MOND regime. Recently, various
hybrid MOND dark matter models were proposed, where cold dark matter (CDM)
phenomenology on cosmological scales and MOND phenomenology on galactic scales
share a common origin. Such models typically contain a mode that is directly
coupled to matter (for MOND), but with non-relativistic sound speed (for CDM).
Thus, even non-relativistic objects like stars can emit gravitational Cherenkov
radiation. We calculate a lower bound on the associated energy loss. We use a
controlled approximation that depends only on the MOND regime of these models.
We apply our results to three concrete models: For the original superfluid dark
matter model (SFDM), we rule out a part of the parameter space, including the
most commonly used parameters. For two-field SFDM, we find no constraint since
the matter coupling of the relevant mode is suppressed by mixing. For the
recently-proposed model by Skordis and Z{\l}o\'snik, we find no constraint
since the matter coupling is suppressed in non-static situations.
