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

Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly