  Hawking's black hole area theorem provides a geometric realization of the
second law of thermodynamics and constrains gravitational processes. In this
work we explore a one-parameter extension of this constraint formulated in
terms of the monotonicity properties of R\'enyi entropies. We focus on black
hole mergers in asymptotically AdS space and determine new restrictions which
these R\'enyi second laws impose on the final state. We evaluate the entropic
inequalities starting from the thermodynamic ensembles description of black
hole geometries, and find that for many situations they set more stringent
bounds than those imposed by the area increase theorem.
