Apparent horizons in cosmologically-embedded black holes

We present exact solutions for the cosmological embedding of a broad class of non-singular black holes, demonstrating that these objects exhibit an apparent horizon. The evolution of the latter is analyzed as a function of the cosmological redshift \(z\). We show that its size exceeds that of the event horizon of an isolated black hole and increases monotonically with increasing \(z\). Explicit formulas and numerical results are provided for the specific cases of the Hayward and Fan \(\&\) Wang non-singular black-hole models. Furthermore, we explore the distinct dynamical roles of the event and apparent horizons, highlighting connection between the latter and the recently identified cosmological mass shift in non-singular black holes.