  The decays $K^{\ast}(892)\rightarrow K_{S,L}^{0}\pi$ can be used to study CP
violation and CPT violation. The $K^{\ast}(892)$ (hereinafter referred to as
$K^{\ast}$) meson can be produced via $J/\psi$ decays at BESIII. In this paper,
we study CP violation and $K_S^0-K_L^0$ asymmetry in the $J/\psi$ decays
involving the $K^{\ast}$ meson in the final states and obtain the following
results \begin{align} {\mathcal A}_{CP}\left(J/\psi\rightarrow
K^{\ast}f_{J/\psi}\rightarrow K^0_{S}\pi
f_{J/\psi}\right)=\left(3.64\pm0.04\right)\times 10^{-3},\nonumber\\ {\mathcal
A}_{CP}\left(J/\psi\rightarrow K^{\ast}f_{J/\psi}\rightarrow K^0_{L}\pi
f_{J/\psi}\right)=\left(-3.32\pm0.04\right)\times 10^{-3},\nonumber \end{align}
and \begin{align} &R\left(J/\psi\rightarrow K^{\ast}f_{J/\psi}\rightarrow
K^0_{S,L}\pi f_{J/\psi} \right)=\left(3.51\pm0.03\right)\times
10^{-3},\nonumber\\ &R\left(J/\psi\rightarrow \bar{K}^{\ast}
\bar{f}_{J/\psi}\rightarrow K^0_{S,L}\pi \bar{f}_{J/\psi}
\right)=\left(-3.45\pm0.03\right)\times 10^{-3}.\nonumber \end{align} Basing on
two cases: the samples of $10^{10}$ and $10^{12}$ $J/\psi$ events, we calculate
the expected numbers of the observed signal events on the CP violation and the
$K_S^0-K_L^0$ asymmetry in the $J/\psi$ decays with $K^{\ast}$ meson in the
final states, we find that the BESIII experiment may be able to unambiguously
observe CP violation and $K_S^0-K_L^0$ asymmetry for each of these two cases.
We study the possibility to constraint the CPT violation parameter $z$ and
discuss the sensitivity for the measurement of $z$ in $J/\psi$ decays with
$K^{\ast}$ meson in the final states at BESIII. The sensitivity of the
measurement of $z$ depend on the measured precision of the parameters
$m_L-m_S$, $\Gamma_L$, $\Gamma_S$, $p$ and $q$ and the consistence between the
values of $t_0$ and $t_1$ and the event selection criteria in experiment.
