While driven interacting quantum matter is generically subject to heating and scrambling, certain classes of systems evade this paradigm. We study such an exceptional class in periodically driven critical (1 + 1)-dimensional systems with a spatially modulated but disorder-free time evolution operator. Instead of complete scrambling, the excitations of the system remain well defined. Their propagation is analogous to the evolution along light cones in a curved spacetime obtained by two black holes. The Hawking temperature serves as an order parameter which distinguishes between heating and nonheating phases. Beyond a timescale determined by the inverse Hawking temperature, excitations are absorbed by the black holes resulting in a singular concentration of energy at their horizon. We obtain these results analytically within conformal field theory, capitalizing on a mapping to sine-squared deformed field theories. Furthermore, by means of numerical calculations for an interacting XXZ spin-¹/₂ chain, we demonstrate that our findings survive lattice regularization.