  The cosmological principle has been verified using electromagnetic (EM)
observations. However its verification with high accuracy is challenging due to
various foregrounds and selection effects, and possible violation of the
cosmological principle has been reported in the literature. In contrast,
gravitational wave (GW) observations are free of these foregrounds and related
selection biases. This may enable future GW experiments to test the
cosmological principle robustly with full sky distribution of millions of
standard bright/dark sirens. However, the sensitivities of GW detectors are
highly anisotropic, resulting in significant instrument induced anisotropies in
the observed GW catalog. We investigate these instrumental effects for 3rd
generation detector networks in term of multipoles $a_{\ell m}$ of the observed
GW source distribution, using Monte Carlo simulations. (1) We find that the
instrument induced anisotropy primarily exists at the $m=0$ modes on large
scales ($\ell \lesssim 10$), with amplitude $\langle |a_{\ell 0}|^2 \rangle
\sim 10^{-3}$ for two detectors (ET-CE) and $\sim 10^{-4}$ for three detectors
(ET-2CE). This anisotropy is correlated with the sky distribution of
signal-to-noise ratio (SNR) and localization accuracy. Such anisotropy sets a
lower limit on the detectable cosmological $a_{\ell 0}$. (2) However, we find
that the instrument induced anisotropy is efficiently canceled by rotation of
the Earth in $m\neq 0$ components of $a_{\ell m}$. Therefore $a_{\ell m}$
($m\neq 0$) are clean windows to detect cosmological anisotropies. (3) We
investigate the capability of 3rd generation GW experiments to measure the
cosmic dipole. Through Monte Carlo simulations, we find that cosmic dipole with
an amplitude of $\sim 10^{-2}$ reported in the literature can be detected/ruled
out by ET-CE and ET-2CE robustly, through the measurement of $a_{11}$.
