We show the presence of universal features in the entanglement entropy of boundary states for conformal field theories on a circle when the reduced density matrix is obtained by tracing over right/left-moving modes. We derive a general forumla for the left-right entanglement entropy in terms of the central charge and the modular S matrix of the theory. For a specific choice of boundary condition, the finite contribution to this measure of entanglement can be fully formulated in terms of the quantum dimensions of the CFT. Furthermore, the form of this finite piece is identical to the average topological entanglemtn entropy in a (2+1)-d TQFT.