Speaking of shrapnel and the relative velocities thereof, there was some discussion of mines earlier... Why use mines when you can use rocks? Put a nice big asteroid in whatever orbital plane you want to render unusable and go at it with a jackhammer for a few weeks, scattering shards of rock into slightly different orbits that will soon form a ring. Better yet, do this in 3 orthogonal orbits, inducing Kessler Syndrome in a way that should eventually create a sphere of gravel in varying and unpredictable orbits equally covering the planet. Put it far enough out and it will take more material, but will mostly spare GPS satellites and space telescopes, the infrastructure needed to look out for incoming enemies.
Granted, thick enough armor will protect against this defense, but it could probably deter lighter craft, missiles, and civilian craft from the planet itself. It might be more useful as a siege tactic if constructed quickly enough, destroying the defenders' orbital infrastructure in one sweep and putting any new launches at risk.
With revolving weapons, maybe better materials science would allow the barrels themselves to have internal channels, with coolant pumped in through a central valve and drained from an outer ring after zigzagging along the barrels. But then you'd need rotating gaskets... Maybe the rotation would help the coolant stay in the system even with an imperfect seal, if it "pools" every time it's transferred outward.