We investigate the thermodynamic phase behavior of rotating and slowly accelerating anti-de Sitter black holes. While we find some similarities with the nonrotating charged counterparts, such as the peculiar phenomena of "snapping swallowtails" we also find subtle but significant distinctions that can be attributed to a qualitatively modified parameter space of the solution. Consequently the zeroth-order phase transition now occurs over a range of pressures, mini-entropic black holes no longer exist in the regime of slow acceleration, and the "no black hole region" emerges continuously—from a zero temperature extremal black hole. The formerly equal transition pressure now experiences a fine splitting, as the emergence of a no black hole region and the zeroth-order phase transition appear at different pressures, different also from the termination pressure of the first order phase transition. This has the further effect of admitting reentrant phase transitions that can be achieved in two ways, either by varying the temperature at fixed pressure or varying the pressure at fixed temperature.