A host of dark energy models and non-standard cosmologies predict an enhanced Hubble rate in the early Universe: perfectly viable models, which satisfy Big Bang Nucleosynthesis (BBN), cosmic microwave background and general relativity tests, may nevertheless lead to enhancements of the Hubble rate up to many orders of magnitude. In this paper we show that strong bounds on the pre-BBN evolution of the Universe may be derived, under the assumption that dark matter is a thermal relic, by combining the dark matter relic density bound with constraints coming from the production of cosmic-ray antiprotons by dark matter annihilation in the Galaxy. The limits we derive can be sizable and apply to the Hubble rate around the temperature of dark matter decoupling. For dark matter masses lighter than 100 GeV, the bound on the Hubble-rate enhancement ranges from a factor of a few to a factor of 30, depending on the actual cosmological model, while for a mass of 500 GeV the bound falls in the range 50-500. Uncertainties in the derivation of the bounds and situations where the bounds become looser are discussed. We finally discuss how these limits apply to some specific realizations of non-standard cosmologies: a scalar-tensor gravity model, kination models and a Randall-Sundrum D-brane model.
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