  Topological phases are unique states of matter which support non-local
excitations which behave as particles with fractional statistics. A universal
characterization of gapped topological phases is provided by the topological
entanglement entropy (TEE). We study the finite size corrections to the TEE by
focusing on systems with $Z_2$ topological ordered state using density-matrix
renormalization group (DMRG) and perturbative series expansions. We find that
extrapolations of the TEE based on the Renyi entropies with Renyi index $n\geq
2$ suffer from much larger finite size corrections than do extrapolations based
on the von Neumann entropy. In particular, when the circumference of the
cylinder is about ten times the correlation length, the TEE obtained using von
Neumann entropy has an error of order $10^{-3}$, while for Renyi entropies it
can even exceed 40%. We discuss the relevance of these findings to previous and
future searches for topological ordered phases, including quantum spin liquids.
