Neutron star mergers provide a unique probe of the dense matter equation of state (EOS) across a wide range of parameter space, from the cold and equilibrated matter of the inspiral, to the shock-heated and higher-density conditions that govern the post-merger evolution. In this talk, I will start with an overview of what we have learned about the EOS so far from the first LIGO-Virgo observations of binary neutron star inspirals, and what we might be able to learn with upcoming detections in the next few years. I will then present a series of neutron star merger simulations, that employ a phenomenological framework to study new parts of the EOS parameter space. Using these simulations, I will highlight some of the additional EOS constraints that we might be able to extract from a future measurement of gravitational waves emitted by the hot and massive remnant neutron star that forms after the merger. These post-merger gravitational waves probe very different conditions than the inspiral, providing the possibility of new constraints on the finite-temperature EOS.