In this work we investigate the radiatively induced Chern-Simons-like terms in four-dimensions at zero and finite temperature. We use the approach of rationalizing the fermion propagator up to the leading order in the CPT-violating coupling b µ . In this approach, we have shown that although the coefficient of Chern-Simons term can be found unambiguously in different regularization schemes at zero or finite temperature, it remains undetermined. We observe a correspondence among results obtained at finite and zero temperature.

We investigate the radiatively induced Chern-Simons-like term in fourdimensional field theory at finite temperature. The Chern-Simons-like term is temperature dependent and breaks the Lorentz and CPT symmetries. We find that this term remains undetermined although it can be found unambiguously in different regularization schemes at finite temperature.

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We verify the consistency of the Gödel-type solutions within the four-dimensional ChernSimons modified gravity with the non-dynamical Chern-Simons coefficient, for different forms of matter including dust, fluid, scalar field and electromagnetic field, and discuss the related causality issues. We show that, unlike the general relativity, a vacuum solution is possible in our theory. Another essentially new result of our theory having no analogue in the general relativity consists in the existence of the hyperbolic causal solutions for a physically wellmotivated matter.