  We report the first measurements of the absolute branching fractions of
$D^0\to K_L^0\phi$, $D^0\to K_L^0\eta$, $D^0\to K_L^0\omega$, and $D^0\to
K_L^0\eta^{\prime}$, obtained by analyzing $2.93\,\rm fb^{-1}$ of $e^+e^-$
collision data taken at a center-of-mass energy of 3.773 GeV with the BESIII
detector. Taking the world averages of the branching fractions of $D^0\to
K_S^0\phi$, $D^0\to K_S^0\eta$, $D^0\to K_S^0\omega$, and $D^0\to
K_S^0\eta^{\prime}$, the $K_S^0$-$K_L^0$ asymmetry $\mathcal{R}(D^0)$ in these
decay modes are obtained. The CP asymmetries in these decays are also
determined. No significant $CP$ violation is observed.
