  We revisit the stability of black hole saddles for the Euclidean path
integral describing the canonical partition function $Z(\beta)$ for gravity
inside a spherical reflecting cavity. The boundary condition at the cavity wall
couples the transverse-traceless (TT) and pure-trace modes that are
traditionally used to describe fluctuations about Euclidean Schwarzschild black
holes in infinite-volume asymptotically flat and asymototically AdS spacetimes.
This coupling obstructs the familiar Gibbons-Hawking-Perry treatment of the
conformal factor problem, as Wick rotation of the pure-trace modes would
require that the TT modes be rotated as well. The coupling also leads to
complex eigenvalues for the \L operator. We nevertheless find that the \L
operator can be diagonalized in the space of coupled modes. This observation
allows the eigenmodes to define a natural generalization of the pure-trace
Wick-rotation recipe used in infinite volume, with the result that a mode with
eigenvalue $\lambda$ is stable when ${\rm Re}\,\lambda > 0$. In any cavity, and
with any cosmological constant $\Lambda \le 0$, we show this recipe to
reproduce the expectation from black hole thermodynamics that large Euclidean
black holes define stable saddles while the saddles defined by small Euclidean
black holes are unstable.
