  We investigate how the propagation of an astrophysical gravitational wave
background (AGWB) is modified over cosmological volumes when considering
theories beyond general relativity of the type Horndeski gravity. We first
deduce an amplitude correction on the AGWB induced for the presence of a
possible running in the Planck mass. Then, we apply the spectral noise density
from some ground-based interferometers, namely, the Advanced LIGO (aLIGO),
Einstein Telescope (ET) and Cosmic Explore (CE), to evaluate the
signal-to-noise ratio (SNR) as a function of the amplitude of the running of
the Planck mass for two different scenarios. We find that for observation time
period $\gtrsim$ 5 yrs and $\gtrsim$ 1 yr, we can have a significant signal of
the AGWB in the band [1-100] Hz from the ET and CE sensitivity, respectively.
Using Fisher information, we find some forecast bounds, and we deduce
$\lesssim$ 27\% and $\lesssim$ 18\% correction at 1$\sigma$ confidence level on
the amplitude of the running of the Planck mass from ET and CE, respectively.
It is clear that a detection of a AGWB in future can open a new window to probe
the nature of gravity with good accuracy.
