  Early dark energy that relieves Hubble tension leaves a fingerprint in the
primordial stochastic gravitational wave (GW) background that originates from
cosmic string network. The signal is not only detectable with future planned GW
experiments, but also distinguishable from other astrophysical and cosmological
signals in the GW frequency spectrum. We find that the cosmic string GW
spectrum can probe other new physics that influence the universe in
post-Big-Bang-Nucleosynthesis with mid-band GW detection, which extends GW
cosmic archaeology search region.
