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 $\mathrm{\&}$ 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.