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The IceCube neutrino observatory at the geographic South Pole is also a unique lab for high-energy cosmic rays. IceCube consists of a cubic-kilometer optical array deep in the ice that measures Cherenkov light from cosmic-ray muons and from neutrino-induced particle cascades. This deep optical array is complemented by IceCube’s surface array, IceTop, comprised of ice-Cherenkov tanks which detect the electromagnetic particles and low-energy muons of cosmic-ray air showers. In addition to serving as a veto, IceTop also makes important contributions to the particle and astrophysics of Galactic cosmic rays in the energy range below 1 EeV. IceCube-Gen2 will be an extension of IceCube targeting astrophysical high-energy neutrinos as primary science case. For this purpose, the volume of the optical array will be increased by an order of magnitude, and an in-ice radio array for ultra-high-energy neutrinos will be added. Following the success of IceCube, also for IceCube-Gen2 a surface array is planned above the footprint of the deep optical array. Through a combination of elevated scintillation detectors and radio antennas, the accuracy for air-shower measurements will be maximized in the energy range of the most energetic Galactic cosmic rays and the presumed transition to extragalactic cosmic rays. Moreover, a thirtyfold increase in the aperture of surface-deep coincident events will enable new ways to study the forward hadronic physics of cosmic-ray air showers. This talk will provide an overview of the design of the IceCube-Gen2 surface array and its science case.