Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly

The CPT odd contribution to the effective electromagnetic action deriving from the vacuum polarization tensor in a large class of fermionic systems exhibiting Lorentz invariance violation (LIV) is calculated using thermal field theory methods, focusing upon corrections depending on the chemical potential. The systems considered exhibit the axial anomaly and their effective actions are described by axion electrodynamics whereby all the LIV parameters enter in the coupling $\Theta(x)$ to the unmodified Pontryagin density. A preliminary application to type-I tilted Weyl semimetals is briefly presented.