  In this paper, the following two propositions are proven under the dominant
energy condition for the matter field in the higher-dimensional spherically
symmetric spacetime in Einstein-Gauss-Bonnet gravity in the presence of a
cosmological constant $\Lambda$. First, for $\Lambda\le 0$ and $\alpha \ge 0$
without a fine-tuning to give a unique anti-de Sitter vacuum, where $\alpha$ is
the Gauss-Bonnet coupling constant, vanishing generalized Misner-Sharp mass is
equivalent to the maximally symmetric spacetime. Under the fine-tuning, it is
equivalent to the vacuum class I spacetime. Second, under the fine-tuning with
$\alpha>0$, the asymptotically anti-de Sitter spacetime in the
higher-dimensional Henneaux-Teitelboim sense is only a special class of the
vacuum class I spacetime. The latter means the universal slow fall-off to the
unique anti-de Sitter infinity in the presence of physically reasonable matter.
