  We examine which information on the early cosmological history can be
extracted from the potential measurement by third-generation gravitational-wave
observatories of a stochastic gravitational wave background (SGWB) produced by
cosmic strings. We consider a variety of cosmological scenarios breaking the
scale-invariant properties of the spectrum, such as early long matter or
kination eras, short intermediate matter and inflation periods inside a
radiation era, and their specific signatures on the SGWB. This requires to go
beyond the usually-assumed scaling regime, to take into account the transient
effects during the change of equation of state of the universe. We compute the
time evolution of the string network parameters and thus the loop-production
efficiency during the transient regime, and derive the corresponding shift in
the turning-point frequency. We consider the impact of particle production on
the gravitational-wave emission by loops. We estimate the reach of future
interferometers LISA, BBO, DECIGO, ET and CE and radio telescope SKA to probe
the new physics energy scale at which the universe has experienced changes in
its expansion history. We find that a given interferometer may be sensitive to
very different energy scales, depending on the nature and duration of the
non-standard era, and the value of the string tension. It is fascinating that
by exploiting the data from different GW observatories associated with distinct
frequency bands, we may be able to reconstruct the full spectrum and therefore
extract the values of fundamental physics parameters.
