  The local position invariance (LPI) is one of the three major pillars of
Einstein equivalence principle, ensuring the space-time independence on the
outcomes of local experiments. The LPI has been tested by measuring the
gravitational redshift effect in various depths of gravitational potentials. We
propose a new cosmological test of the LPI by observing the asymmetry in the
cross-correlation function between different types of galaxies, which
predominantly arises from the gravitational redshift effect induced by the
gravitational potential of halos at which the galaxies reside. We show that the
ongoing and upcoming galaxy surveys can give a fruitful constraint on the
LPI-violating parameter, $\alpha$, at distant universes (redshift
$z\sim0.1-1.8$) over the cosmological scales (separation $s\sim5-10\, {\rm
Mpc}/h$) that have not yet been explored, finding that the expected upper limit
on $\alpha$ can reach $0.03$.
