The excitation of cavity standing waves in double-slit structures in thin gold films, with slit lengths between 400 and 2560 nm, was probed with a strongly focused electron beam in a transmission electron microscope. The energies and wavelengths of cavity modes up to the $11\mathrm{th}$ mode order were measured with electron energy loss spectroscopy to derive the corresponding dispersion relation. For all orders, a significant redshift of mode energies accompanied by a wavelength elongation relative to the expected resonator energies and wavelengths is observed. The resultant dispersion relation is found to closely follow the well-known dispersion law of surface-plasmon polaritons (SPPs) propagating on a gold/air interface, thus providing direct evidence for the hybridized nature of the detected cavity modes with SPPs.