"Stability" is generally taken to be the justification for requiring that the spectrum of the Hamiltonian should be bounded below. The spectrum of the Hamiltonian is not bounded below for thermal sectors, however, but thermal states are nonetheless taken to be stable because they satisfy thermodynamic constraints. In classical Physics, we would say that the thermal state has lowest free energy, which is a thermodynamic concept distinct from Hamiltonian operators that generate time-like translations.
The entropy component of free energy, meanwhile, is a nonlinear functional of the quantum state (presuming that the definition of entropy in quantum field theory would be at least this much like von Neumann's definition in terms of density operators), so we can reasonably expect the sum of the energy and entropy components to have a minimum in the state of greatest symmetry, as we see for thermal states. [It seems particularly notable in this context that the entropy is not an observable in the usual quantum mechanical sense of a linear functional of the quantum state.]
The presence of irreducible randomness in quantum mechanics presumably puts quantum field theory as much in the conceptual space of thermodynamics as in the conceptual space of classical mechanics, despite the quasi-functorial relationship of "quantization", so perhaps we should expect there to be some relevance of thermodynamic concepts.
Given this background (assuming, indeed, that no part of it is too tendentious), why should we think that requiring the Hamiltonian to have a spectrum that is bounded below should have anything to do with stability in the case of an interacting field? The fact that we can construct a vacuum sector for free fields in which the spectrum of the Hamiltonian is bounded below does not seem enough justification for interacting fields that introduce nontrivial biases towards statistically more complex states.
This question is partly motivated by John Baez' discussion of "quantropy" on Azimuth. I am also interested in the idea that if we release ourselves from the requirement that the Hamiltonian of interacting fields must have a spectrum that is bounded below, then we will have to look for analogues of the KMS condition for thermal states that restore some kind of analytic structure for interacting fields.
I asked a related Question here, almost a year ago. I don't see an answer to the present Question in the citations given in Tim van Beek's Answer there.