  We make forecasts for the impact a future "midband" space-based gravitational
wave experiment, most sensitive to $10^{-2}- 10$ Hz, could have on potential
detections of cosmological stochastic gravitational wave backgrounds (SGWBs).
Specific proposed midband experiments considered are TianGo, B-DECIGO and
AEDGE. We propose a combined power-law integrated sensitivity (CPLS) curve
combining GW experiments over different frequency bands, which shows the
midband improves sensitivity to SGWBs by up to two orders of magnitude at
$10^{-2} - 10$ Hz. We consider GW emission from cosmic strings and phase
transitions as benchmark examples of cosmological SGWBs. We explicitly model
various astrophysical SGWB sources, most importantly from unresolved black hole
mergers. Using Markov Chain Monte Carlo, we demonstrated that midband
experiments can, when combined with LIGO A+ and LISA, significantly improve
sensitivities to cosmological SGWBs and better separate them from astrophysical
SGWBs. In particular, we forecast that a midband experiment improves
sensitivity to cosmic string tension $G\mu$ by up to a factor of $10$, driven
by improved component separation from astrophysical sources. For phase
transitions, a midband experiment can detect signals peaking at $0.1 - 1$ Hz,
which for our fiducial model corresponds to early Universe temperatures of
$T_*\sim 10^4 - 10^6$ GeV, generally beyond the reach of LIGO and LISA. The
midband closes an energy gap and better captures characteristic spectral shape
information. It thus substantially improves measurement of the properties of
phase transitions at lower energies of $T_* \sim O(10^3)$ GeV, potentially
relevant to new physics at the electroweak scale, whereas in this energy range
LISA alone will detect an excess but not effectively measure the phase
transition parameters. Our modelling code and chains are publicly available.
