In a spacetime with a bifurcate Killing horizon, one can use the Killing vector on one side of the horizon to prepare a smooth global state across it, by performing a Euclidean path integral to the Hawking temperature. Viewed in terms of the Hamiltonian that Lie drags fields between slices of constant Killing time, this state will thus be a thermal state at the Hawking temperature.
In a maximally symmetric spacetime such as Minkowski spacetime, this state will also be a stationary state at zero temperature of the Hamiltonian of the global timelike Killing vector. I.e. the Minkowski vacuum is a thermal state at the Unruh temperature of a Rindler Hamiltonian.
However in general there will be no second timelike Killing vector. For example, in the most interesting case of Schwarzschild spacetime, the Hawking Hartle vacuum is a thermal state at the Hawking temperature of the Schwarzschild Hamiltonian. This state is supposed to be "positive frequency with respect to the Kruskal observers"; e.g. their appropriately-renormalized Unruh-DeWitt detectors are not excited.
However, the Kruskal Hamiltonian has an explicit time dependence. It is therefore not clear what it would mean for the Hawking-Hartle vacuum to be its "ground state". How does the Hawking-Hartle vacuum relate to it?