  We investigate static and dynamical n(\ge 6)-dimensional black holes in
Einstein-Gauss-Bonnet gravity of which horizons have the isometries of an
(n-2)-dimensional Einstein space with a condition on its Weyl tensor originally
given by Dotti and Gleiser. Defining a generalized Misner-Sharp quasi-local
mass that satisfies the unified first law, we show that most of the properties
of the quasi-local mass and the trapping horizon are shared with the case with
horizons of constant curvature. It is shown that the Dotti-Gleiser solution is
the unique vacuum solution if the warp factor on the (n-2)-dimensional Einstein
space is non-constant. The quasi-local mass becomes constant for the
Dotti-Gleiser black hole and satisfies the first law of the black-hole
thermodynamics with its Wald entropy. In the non-negative curvature case with
positive Gauss-Bonnet constant and zero cosmological constant, it is shown that
the Dotti-Gleiser black hole is thermodynamically unstable. Even if it becomes
locally stable for the non-zero cosmological constant, it cannot be globally
stable for the positive cosmological constant.
