We revisit the flavor composition of high-energy astrophysical neutrinos observed at neutrino telescopes. Assuming unitary time evolution of the neutrino flavor states, the flavor composition observable at Earth is related to the initial composition at their sources via oscillation-averaged flavor transitions. In a previous study we derived general bounds on the flavor composition of TeV-PeV astrophysical neutrinos assuming three-flavor unitary mixing. We extend these bounds to the case of active-sterile neutrino mixing. Our bounds are analytical, derived based only on the unitarity of the mixing, and do not require sampling over the values of the unknown active-sterile mixing parameters. These bounds apply to any extended active-sterile neutrino mixing scenario where energy-dependent nonstandard flavor mixing dominates over the standard mixing observed in accelerator, reactor, and atmospheric neutrino oscillations.