  We study the quantum Fisher information (QFI) and, thus, the multipartite
entanglement structure of thermal pure states in the context of the Eigenstate
Thermalization Hypothesis (ETH). In both the canonical ensemble and the ETH,
the quantum Fisher information may be explicitly calculated from the response
functions. In the case of ETH, we find that the expression of the QFI bounds
the corresponding canonical expression from above. This implies that although
average values and fluctuations of local observables are indistinguishable from
their canonical counterpart, the entanglement structure of the state is starkly
different; with the difference amplified, e.g., in the proximity of a thermal
phase transition. We also provide a state-of-the-art numerical example of a
situation where the quantum Fisher information in a quantum many-body system is
extensive while the corresponding quantity in the canonical ensemble vanishes.
Our findings have direct relevance for the entanglement structure in the
asymptotic states of quenched many-body dynamics.
