  The recently proposed Skordis-Z\l{}o\'{s}nik theory is the first relativistic
MOND theory that can recover the success of the standard $\Lambda$CDM model at
matching observations of the cosmic microwave background. This paper aims to
revisit the Newtonian and MOND approximations and the gravitational wave
analysis of the theory. For the local gravitational parameters, we show that
one could obtain both time-varying effective Newtonian gravitational
\textit{constant} $G_\textrm{N}$ and time-varying characteristic MOND
acceleration scale $a_\textrm{MOND}$, by relaxing the static assumption
extensively adopted in the literature. Specially, we successfully demonstrate
how to reproduce the redshift dependence of $a_\textrm{MOND}$ observed in the
\textit{Magneticum} cold dark matter simulations. For the gravitational waves,
we show that there are only two tensor polarizations, and reconfirm that its
speed is equal to the speed of light.
