  If Einstein-Gauss-Bonnet gravity is obtained as a low energy limit of string
theory, then the Gauss-Bonnet parameter $\alpha$ is essentially the inverse
string tension and thus necessarily positive. If one treats
Einstein-Gauss-Bonnet gravity as a modified theory of gravity in the anti-de
Sitter bulk in the bottom-up approach of holography, then there is no obvious
restriction on the sign of the parameter a priori, though various studies
involving boundary causality have restricted the possible range of $\alpha$. In
this short note, we argue that if holographic descriptions are to be
consistent, then the Gauss-Bonnet parameter has to be positive. This follows
from a geometric consistency condition in the Euclidean picture. From the
Lorentzian signature perspective, black holes with a negative $\alpha$ lead to
uncontrolled brane nucleation in the bulk and so the supposedly static geometry
is untenable. In fact, even the ground state without a black hole is
problematic. In other words, the bottom-up approach agrees with the top-down
approach on the sign of the parameter. Some possible loopholes of the
conclusion are discussed.
