We study the near threshold behavior of cross sections of low-energy antiproton scattering off the ground and excited states of positronium with zero total orbital momentum L = 0. In our computational experiment, the existence of singularities called the Gailitis–Damburg oscillations above the thresholds of excited states of positronium and antihydrogen atoms is confirmed. In the future the obtained results can be useful for devel- oping proposals for improving the conditions of experiments with antimatter.