  We investigate entanglement properties in the ground state of the
open/periodic SU($n$) generalized valence-bond-solid state consisting of
representations of SU($n$). We obtain exact expression for the reduced density
matrix of a block of contiguous spins and explicitly evaluate the von Neumann
and the R\'enyi entropies. We discover that the R\'enyi entropy is independent
of the parameter $\alpha$ in the limit of large block sizes and its value $2
\log n$ coincides with that of von Neumann entropy. We also find the direct
relation between the reduced density matrix of the subsystem and edge states
for the corresponding open boundary system.
