We show that uniformly accelerated detectors can display genuinely thermal features even if the Kubo-Martin-Schwinger (KMS) condition fails to hold. These features include satisfying thermal detailed balance and having a Planckian response identical to cases in which the KMS condition is satisfied. In this context, we discuss that satisfying the KMS condition for accelerated trajectories is just sufficient but not necessary for the Unruh effect to be present in a given quantum field theory. Furthermore, we extract the necessary and sufficient conditions for the response function of an accelerated detector to be thermal in the infinitely adiabatic limit. This analysis provides new insight about the interplay between the KMS condition and the Unruh effect, and a solid framework in which the robustness of the Unruh effect against deformations of quantum field theories (perhaps Lorentz-violating) can be answered unambiguously.