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We investigate the R\'enyi entropy and entanglement entropy of an intervalwith an arbitrary length in the canonical ensemble, microcanonical ensemble andprimary excited states at large energy density in the thermodynamic limit of atwo-dimensional large central charge $c$ conformal field theory. As ageneralization of the recent work [Phys. Rev. Lett. 122 (2019) 041602], themain purpose of the paper is to see whether one can distinguish these variouslarge energy density states by the R\'enyi entropies of an interval atdifferent size scales, namely, short, medium and long. Collecting earlierresults and performing new calculations in order to compare with and fill gapsin the literature, we give a more complete and detailed analysis of theproblem. Especially, we find some corrections to the recent results for theholographic R\'enyi entropy of a medium size interval, which enlarge thevalidity region of the results. Based on the R\'enyi entropies of the threeinterval scales, we find that R\'enyi entropy cannot distinguish the canonicaland microcanonical ensemble states for a short interval, but can do the job forboth medium and long intervals. At the leading order of large $c$ theentanglement entropy cannot distinguish the canonical and microcanonicalensemble states for all interval lengths, but the difference of entanglemententropy for a long interval between the two states would appear with $1/c$corrections. We also discuss R\'enyi entropy and entanglement entropydifferences between the thermal states and primary excited state. Overall, ourwork provides an up-to-date picture of distinguishing different thermal orprimary states at various length scales of the subsystem.

R\'enyi entropy at large energy density in 2D CFT