  We investigate the R\'enyi entropy and entanglement entropy of an interval
with an arbitrary length in the canonical ensemble, microcanonical ensemble and
primary excited states at large energy density in the thermodynamic limit of a
two-dimensional large central charge $c$ conformal field theory. As a
generalization of the recent work [Phys. Rev. Lett. 122 (2019) 041602], the
main purpose of the paper is to see whether one can distinguish these various
large energy density states by the R\'enyi entropies of an interval at
different size scales, namely, short, medium and long. Collecting earlier
results and performing new calculations in order to compare with and fill gaps
in the literature, we give a more complete and detailed analysis of the
problem. Especially, we find some corrections to the recent results for the
holographic R\'enyi entropy of a medium size interval, which enlarge the
validity region of the results. Based on the R\'enyi entropies of the three
interval scales, we find that R\'enyi entropy cannot distinguish the canonical
and microcanonical ensemble states for a short interval, but can do the job for
both medium and long intervals. At the leading order of large $c$ the
entanglement entropy cannot distinguish the canonical and microcanonical
ensemble states for all interval lengths, but the difference of entanglement
entropy for a long interval between the two states would appear with $1/c$
corrections. We also discuss R\'enyi entropy and entanglement entropy
differences between the thermal states and primary excited state. Overall, our
work provides an up-to-date picture of distinguishing different thermal or
primary states at various length scales of the subsystem.
