I suspect the OP may also be concerned because minocycline crosses the blood-brain barrier, so it's reasonable to expect that the FAA may have an issue with it. The AOPA database does say it's allowed with status report and no side effects. Generally, meds that are allowed under those criteria don't require you to stand down beyond a week or two to make sure you don't have side-effects, but I'd ask Doc Bruce about this one because of its unusual penetration.
As Ron says, it probably also depends on why it's being prescribed. From Wikipedia, it's commonly prescribed for acne, but it has other uses too, including Lyme Disease, which is probably grounding.