  We evaluate one-interval R\'enyi entropy and entanglement entropy for the
excited states of two-dimensional conformal field theory (CFT) on a cylinder,
and examine their differences from the ones for the thermal state. We assume
the interval to be short so that we can use operator product expansion (OPE) of
twist operators to calculate R\'enyi entropy in terms of sum of one-point
functions of OPE blocks. We find that the entanglement entropy for highly
excited state and thermal state behave the same way after appropriate
identification of the conformal weight of the state with the temperature.
However, there exists no such universal identification for the R\'enyi entropy
in the short-interval expansion. Therefore, the highly excited state does not
look thermal when comparing its R\'enyi entropy to the thermal state one. As
the R\'enyi entropy captures the higher moments of the reduced density matrix
but the entanglement entropy only the average, our results imply that the
emergence of thermality depends on how refined we look into the entanglement
structure of the underlying pure excited state.
