  Nonlocal RT gravity is a successful modified gravity theory, which not only
explains the late-time cosmic acceleration but also behaves well in the solar
system. Previous analysis generally assumes the auxiliary field $S_i$ vanishes
at the cosmic background. However, we find the background $S_i$ is proportional
to $a^2$ with the expansion of the universe. Then we discuss the influence of
the nonzero background $S_i$ on the cosmic background evolution, the scalar and
tensor perturbations. We find the cosmic background evolution is independent of
$S_i$, and the influence of the nonzero background $S_i$ on the weak field
limit at solar system scales is negligible. For the tensor perturbation, we
find the only possible observable effect is the influence of nonzero background
$S_i$ on the LIGO gravitational wave amplitude and also luminosity distance.
Future high redshift gravitational wave observations could be used to constrain
the background value of $S_i$.
