We present the design of the Fabry-Pérot system of the Robert Stobie Spectrograph on the 10 m class Southern African Large Telescope and its characterization as measured in the laboratory. This system provides spectroscopic imaging at any desired wavelength spanning a bandpass 430-860 nm, at four different spectral resolving powers (λ/∆λ) ranging from 300 to 9000. Our laboratory tests revealed a wavelength dependence of the etalon gap and parallelism with a maximum variation between 600-720 nm that arises because of the complex structure of the broadband multi-layer dielectric coatings. We also report an unanticipated optical effect of this multi-layer coating structure that produces a significant, and wavelength-dependent, change in the apparent shape of the etalon plates. This change is caused by two effects: the physical non-uniformities or thickness variations in the coating layers, and the wavelength dependence of the phase change upon reflection that can amplify these non-uniformities. We discuss the impact of these coating effects on the resolving power, finesse, and throughput of the system. This Fabry-Pérot system will provide a powerful tool for imaging spectroscopy on one of the world's largest telescopes.