Measuring the flavor composition of the TeV-PeV astrophysical neutrinos, i.e., the ratio of the flux of neutrinos of each flavor to the total flux sheds light on their production mechanisms and on the action of flavor transitions during propagation. So far, measurements of the flavor composition, based on IceCube data, have of necessity assumed that it is independent of neutrino energy, on account of the limited size of the data sample. However, the natural expectation is for the flavor composition to vary with neutrino energy, due to the presence of different neutrino production mechanisms at different energies, or to flavor-changing new physics. Therefore, we look for signs of the energy dependence of the flavor composition in recent IceCube public data and show forecasts for next-generation neutrino telescopes: Baikal-GVD, IceCube-Gen2, KM3NeT, P-ONE, TAMBO, and TRIDENT. We find that combing the data samples of telescopes in the future is critical to pin down changes in the flavor composition with energy.