  The existence of black holes is one of the key predictions of general
relativity (GR) and therefore a basic consistency test for modified theories of
gravity. In the case of spherical symmetry in GR the existence of an apparent
horizon and its regularity is consistent with only two distinct classes of
physical black holes. Here we derive constraints that any self-consistent
modified theory of gravity must satisfy to be compatible with their existence.
We analyze their properties and illustrate characteristic features using the
Starobinsky model. Both of the GR solutions can be regarded as zeroth-order
terms in perturbative solutions of this model. We also show how to construct
nonperturbative solutions without a well-defined GR limit.
