  Owing to the quadratic nature of the theory, Einstein-Gauss-Bonnet gravity
generically permits two distinct vacuum solutions. One solution (the "Einstein"
vacuum) has a well defined limit as the Gauss-Bonnet coupling goes to zero,
whereas the other solution (the "stringy" vacuum) does not. There has been some
debate regarding the stability of these vacua, most recently from Deser and
Tekin who have argued that the corresponding black hole solutions have positive
mass and as such both vacua are stable. Whilst the statement about the mass is
correct, we argue that the stringy vacuum is still perturbatively unstable.
Simply put, the stringy vacuum suffers from a ghost-like instability that is
not excited by the spherically symmetric black hole, but would be excited by
any source likely to emit gravitational waves, such as a binary system. This
result is reliable except in the strongly coupled regime close to the
Chern-Simons limit, when the two vacua are almost degenerate. In this regime,
we study instanton transitions between branches via bubble nucleation, and
calculate the nucleation probability. This demonstrates that there is large
mixing between the vacua, so that neither of them can accurately describe the
true quantum vacuum. We also present a new gravitational instanton describing
black hole pair production in de Sitter space on the Einstein branch, which is
preferred to the usual Nariai instantons, and is not present in pure Einstein
gravity.
