  In this paper we consider the defocusing Hartree nonlinear Schr\"odinger
equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier
multiplier decaying like $|k|^{-\beta}$. By relying on the method of random
averaging operators in arXiv:1910.08492, we show that there exists $\frac{1}{2}
\ll \beta_0 <1 $ such that for $ \beta > \beta_0 $ we have invariance of the
associated Gibbs measure and global existence of strong solutions in its
statistical ensemble. In this way we extend Bourgain's seminal result [7] which
requires $\beta >2$ in this case.
