We propose a new theory of dark matter based on axion physics and cosmological phase transitions. We show that theories in which a gauge coupling increases through a first-order phase transition naturally result in ‘axion relic pockets’: regions of relic false vacua stabilised by the pressure from a kinematically trapped, hot axion gas. Axion relic pockets provide a viable and highly economical theory of dark matter: the macroscopic properties of the pockets depend only on a single parameter (the phase transition temperature). We describe the formation, evolution and present-day properties of axion relic pockets, and outline how their phenomenology is distinct from existing dark matter paradigms. We briefly discuss how laboratory experiments and astronomical observations can be used to test the theory, and identify gamma-ray observations of magnetised, dark-matter-dense environments as particularly promising.