  We show that there are no general stability results for the logarithmic
Sobolev inequality in terms of the Wasserstein distances and $L^{p}(d\gamma)$
distance for $p>1$. To this end, we construct a sequence of centered
probability measures such that the deficit of the logarithmic Sobolev
inequality converges to zero but the relative entropy and the moments do not,
which leads to instability for the logarithmic Sobolev inequality. As an
application, we prove instability results for Talagrand's transportation
inequality and the Beckner--Hirschman inequality.
