Evidence of "new physics" seems to be lacking (in my research into this question.) Black holes can be described fully by known consequences of general relativity as far as our ability to observe them. There are known deviations from the Scharwchild metric, for example in black holes in which the accretion disc is inclined relative to the spin of the black hole, and in which the effects of frame-dragging can be observed as the accretion disc transtions to the spin plane of the black hole.
High energy cosmic rays are a poorly understood phenonomum. They seem to be charged particles (ie protons) that have been acclerated to very high velocities (much higher than 99% c) by black holes. But the particular mechansim that does this is unclear. A paper, https://arxiv.org/pdf/1904.04654.pdf, proposes that high energy protons may be an consequence of black holes with a Kerr-Newmann metric, that is a rotating charged black hole; as even a small charge would have a singificant effect on protons in the neighbourhood. This could be interpreted as evidence that some black holes have a Kerr-Newman spacetime metric.