In this work we study the time evolution of the Renyi entanglement entropy for locally excited states created by twist operators in the cyclic orbifold (T-2)(n)/Z(n) and the symmetric orbifold (T-2)(n)/S-n. We find that when the square of its compactification radius is rational, the second Renyi entropy approaches a universal constant equal to the logarithm of the quantum dimension of the twist operator. On the other hand, in the non-rational case, we find a new scaling law for the Renyi entropies given by the double logarithm of time log log t for the cyclic orbifold CFT.