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A number of proposed and ongoing experiments search for axion dark matter with a mass nearing the limit set by small scale structure. In this talk, I will discuss the late universe cosmology of these models and argue that requiring the axion to have a matter-power spectrum that matches that of cold dark matter constrains the magnitude of the axion couplings to the visible sector. Comparing these limits to current and future experimental efforts, I will argue that many searches require axions with abnormally large couplings to Standard Model fields, independently of how the axion was populated in the early Universe. I will review model building mechanisms that can potentially alleviate the bounds, such as the introduction of large charges, various forms of kinetic mixing, a clockwork structure, and imposing a discrete symmetry. I will also briefly discuss ongoing work to detect a cosmic axion background (CaB) through the nucleon coupling portal using nuclear magnetic resonance and atomic transitions.