The answer is in the same document, § 2.1 AdS Black Holes , page 4

The thermofield hamiltonian (2.2) generates boosts which are translations of the usual hyperbolic angle $\omega$. One can think of the boost as propagating upward on the right side of the Penrose diagram, and downward on the left. The state (2.1) is an eigenvector of $H_{tf}$ with eigenvalue zero, and is therefore boost invariant.

and then page 37

For any non-trivial $\theta$ these are states that are boost invariant and differ from the Minkowski vacuum.

those above.