  We investigate a weak version of subsystem eigenstate thermalization
hypothesis (ETH) for a two-dimensional large central charge conformal field
theory by comparing the local equivalence of high energy state and thermal
state of canonical ensemble. We evaluate the single-interval R\'enyi entropy
and entanglement entropy for a heavy primary state in short interval expansion.
We verify the results of R\'enyi entropy by two different replica methods. We
find nontrivial results at the eighth order of short interval expansion, which
include an infinite number of higher order terms in the large central charge
expansion. We then evaluate the relative entropy of the reduced density
matrices to measure the difference between the heavy primary state and thermal
state of canonical ensemble, and find that the aforementioned nontrivial eighth
order results make the relative entropy unsuppressed in the large central
charge limit. By using Pinsker's and Fannes-Audenaert inequalities, we can
exploit the results of relative entropy to yield the lower and upper bounds on
trace distance of the excited-state and thermal-state reduced density matrices.
Our results are consistent with subsystem weak ETH, which requires the above
trace distance is of power-law suppression by the large central charge.
However, we are unable to pin down the exponent of power-law suppression. As a
byproduct we also calculate the relative entropy to measure the difference
between the reduced density matrices of two different heavy primary states.
