  We investigate properties of a quasi-local mass in a higher-dimensional
spacetime having symmetries corresponding to the isomertries of an
$(n-2)$-dimensional maximally symmetric space in Einstein-Gauss-Bonnet gravity
in the presence of a cosmological constant. We assume that the Gauss-Bonnet
coupling constant is non-negative. The quasi-local mass was recently defined by
one of the authors as a counterpart of the Misner-Sharp quasi-local mass in
general relativity. The quasi-local mass is found to be a quasi-local conserved
charge associated with a locally conserved current constructed from the
generalized Kodama vector and exhibits the unified first law corresponding to
the energy-balance law. In the asymptotically flat case, it converges to the
Arnowitt-Deser-Misner mass at spacelike infinity, while it does to the
Deser-Tekin and Padilla mass at infinity in the case of asymptotically AdS.
Under the dominant energy condition, we show the monotonicity of the
quasi-local mass for any $k$, while the positivity on an untrapped hypersurface
with a regular center is shown for $k=1$ and for $k=0$ with an additional
condition, where $k=\pm1,0$ is the constant sectional curvature of each spatial
section of equipotential surfaces. Under a special relation between coupling
constants, positivity of the quasi-local mass is shown for any $k$ without
assumptions above. We also classify all the vacuum solutions by utilizing the
generalized Kodama vector. Lastly, several conjectures on further
generalization of the quasi-local mass in Lovelock gravity are proposed.
