  We analyze the semiclassical stability of the Schwarzschild AdS black hole in
the Euclidean partition function approach. We perform this computation in the
large $D$ limit and focus on scalar perturbations. We obtain the equations for
non-spherically symmetric scalar perturbations in a simple form. For a class of
perturbations stability is demonstrated by the S-deformation method. For some
other classes we rule out unstable modes of $\mathcal{O}(D^2)$. We also analyze
the spherically symmetric perturbations and demonstrate the appearance of an
unstable mode for small black holes in the large $D$ limit. We obtain an
expression for the eigenvalue corresponding to the unstable mode to next to
leading order in a $1/D$ expansion. This result agrees with a previously
obtained numerical bound on this eigenvalue. For cosmological constant zero,
our answer matches a previous result obtained for the corresponding eigenvalue
for the $D$ dimensional Schwarzschild-Tangherlini black hole to next to leading
order in a $1/D$ expansion.
