Refractive index sensors based on the localized surface plasmon resonance (LSPR) have emerged as powerful tools in various chemosensing and biosensing applications. However, owing to their limited decay length and strong radiation damping, LSPR sensors always suffer from low sensitivity and small figure of merit (FOM). Here, we fabricate a plasmonic nanocavity sensor consisting of a hexagonal Au nanoplate positioned over an ultrasmooth Au film. The strong coupling between the nanoplate and the lower metal film allows for the formation of a plasmonic gap mode that enhances the interaction of the local field with the ambient glycerol solution to increase the sensitivity. Meanwhile, the plasmonic gap mode has a trait of an antiphase charge oscillation in the gap region, imparting a strongly reduced radiative damping and a subsequently promoted FOM. The performance of our proposed refractive index sensor is further boosted by decreasing the gap size of the nanocavity, yielding an outstanding FOM of 11.2 RIU-1 that is the highest yet reported for LSPR sensing in a single nanostructure.