We construct and study rotating axion boson stars (RABSs). These are the spinning generalisations of the spherical gravitating solitons recently introduced in [1]. RABSs are asymptotically flat, stationary, axially symmetric, everywhere regular solutions of the Einstein-Klein-Gordon theory, in the presence of a periodic scalar potential arising in models of axion-like particles. The potential is characterised by two parameters: the mass of the scalar field ma and the decay constant fa. We present an overview of the solution space, for different values of fa, and analyse some of their phenomenological properties. For large decay constants the solutions become identical to the standard spinning mini boson stars. For small decay constants, on the other hand, the solutions develop distinctive features. In particular, we analyse their compactness, the emergence of ergoregions, light rings and the distribution of stable and unstable equatorial timelike circular orbits, including the innermost stable circular orbit. We also establish the occurrence of violations of the strong energy condition for physical observers, for some RABSs. We observe some analogy between RABSs and spinning gravitating Q-balls.
