Structure of radiatively induced Lorentz and $\mathrm{CPT}$ violation in QED at finite temperature

We obtain the induced Lorentz- and $\mathrm{CPT}$-violating term in QED at finite temperature using the imaginary-time formalism and dimensional regularization. Its form resembles a Chern-Simons-like structure, but, unexpectedly, it does not depend on the temporal component of the fixed $b_{\mu }$ constant vector that is coupled to the axial-vector current. Nevertheless, Ward identities are respected and its coefficient vanishes at $T=0,$ consistent with previous computations with the same regularization procedure, and it is a nontrivial function of temperature. We argue that at finite T a Chern-Simons-like Lorentz- and $\mathrm{CPT}$-violating term is generically present, the value of its coefficient being unambiguously determined up to a T-independent constant, related to the zero-temperature renormalization conditions.