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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 Złośnik, we find no constraint since the matter coupling is suppressed in non-static situations.