  We study the effects of pre-recombination physics on the Stochastic
Gravitational Wave Background (SGWB) anisotropies induced by the propagation of
gravitons through the large-scale density perturbations and their
cross-correlation with Cosmic Microwave Background (CMB) temperature and E-mode
polarization ones. As examples of Early Universe extensions to the $\Lambda$CDM
model, we consider popular models featuring extra relativistic degrees of
freedom, a massless non-minimally coupled scalar field, and an Early Dark
Energy component. Assuming the detection of a SGWB, we perform a Fisher
analysis to assess in a quantitative way the capability of future gravitational
wave interferometers (GWIs) in conjunction with a future large-scale CMB
polarization experiment to constrain such variations. Our results show that the
cross-correlation of CMB and SGWB anisotropies will help tighten the
constraints obtained with CMB alone, with an improvement that significantly
depends on the specific model as well as the maximum angular resolution
$\ell_{\rm max}^{\rm GW}$ of the GWIs, their designed sensitivity, and the
amplitude $A_*$ of the monopole of the SGWB.
