We provide an example to show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}$ distance, for $p>1$. The results imply that the stability bounds for the logarithmic Sobolev inequality with respect to $W_{1}$, $W_{2}$, and $L^{1}$ in the space of probability measures with bounded second moments are best possible. As an application of the example, we prove an instability result for the Beckner--Hirschman inequality in terms of $L^{p}$, for $p>2$.