Future cosmic microwave background experiments have the potential to probe the density of relativistic species at the subpercent level. This sensitivity allows light thermal relics to be detected up to arbitrarily high decoupling temperatures. Conversely, the absence of a detection would require extra light species never to have been in equilibrium with the Standard Model. In this Letter, we exploit this feature to demonstrate the sensitivity of future cosmological observations to the couplings of axions to photons, gluons, and charged fermions. In many cases, the constraints achievable from cosmology will surpass existing bounds from laboratory experiments and astrophysical observations by orders of magnitude.