  We explore various aspects of dynamical black holes defined by a future outer
trapping horizon in $n(\ge 5)$-dimensional Einstein-Gauss-Bonnet gravity. In
the present paper, we assume that the spacetime has symmetries corresponding to
the isometries of an $(n-2)$-dimensional maximally symmetric space and the
Gauss-Bonnet coupling constant is non-negative. Depending on the existence or
absence of the general relativistic limit, solutions are classified into GR and
non-GR branches, respectively. Assuming the null energy condition on matter
fields, we show that a future outer trapping horizon in the GR branch possesses
the same properties as that in general relativity. In contrast, that in the
non-GR branch is shown to be non-spacelike with its area non-increasing into
the future. We can recognize this peculiar behavior to arise from a fact that
the null energy condition necessarily leads to the null convergence condition
for radial null vectors in the GR branch, but not in the non-GR branch. The
energy balance law yields the first law of a trapping horizon, from which we
can read off the entropy of a trapping horizon reproducing Iyer-Wald's
expression. The entropy of a future outer trapping horizon is shown to be
non-decreasing in both branches along its generator.
