The electromagnetic fields which are currently observed in the Universe are characteristic for clearly macroscopic systems and can be analyzed within the framework of Maxwell's theory of electromagnetism. However, we will argue that for a high temperature systems - for instance in the early Universe and young neutron stars - the quantum anomalous effects will introduce important corrections and modification of the theory, leading to significant effects even on macroscopic scales. The presented anomalous electrodynamics is thus a borderline between classical and quantum realms.