Förster resonance energy transfer (FRET) is widely applied in chemistry, biology and nanosciences to assess distances on sub-10 nm scale. Extending the range and applicability of FRET requires enhancement of the fluorescence energy transfer at a spatial scale comparable to the donor-acceptor distances. Plasmonic nanoantennas are ideal to concentrate optical fields at a nanoscale fully matching the FRET distance range. Here, we present a resonant aluminum nanogap antenna tailored to enhance single molecule FRET. A 20 nm gap confines light into a nanoscale volume, providing a field gradient on the scale of the donor-acceptor distance, a large 10-fold increase in the local density of optical states, and strong intensity enhancement. With our dedicated design, we obtain 20-fold enhancement on the fluorescence emission of donor and acceptor dyes, and most importantly up to 5-fold enhancement of the FRET rate for donor-acceptor separations of 10 nm. We also provide a thorough framework of the fluorescence photophysics occurring in the nanoscale gap volume. The presented enhancement of energy transfer flow at the nanoscale opens a yet unexplored facet of the various advantages of optical nanoantennas and provides a new strategy towards biological applications of single molecule FRET at micromolar concentrations.