  The Eigenstate Thermalization Hypothesis (ETH) provides a way to understand
how an isolated quantum mechanical system can be approximated by a thermal
density matrix. We find a class of operators in (1+1)-$d$ conformal field
theories, consisting of quasi-primaries of the identity module, which satisfy
the hypothesis only at the leading order in large central charge. In the
context of subsystem ETH, this plays a role in the deviation of the reduced
density matrices, corresponding to a finite energy density eigenstate from its
hypothesized thermal approximation. The universal deviation in terms of the
square of the trace-square distance goes as the 8th power of the subsystem
fraction and is suppressed by powers of inverse central charge ($c$).
Furthermore, the non-universal deviations from subsystem ETH are found to be
proportional to the heavy-light-heavy structure constants which are typically
exponentially suppressed in $\sqrt{h/c}$, where $h$ is the conformal scaling
dimension of the finite energy density state. We also examine the effects of
the leading finite size corrections.
