Just wondering if anyone on here is running a AMR500 blower and could give me a rough idea of how hot the casing should get. Can't find much info at all online about blower case temperatures.

Guessing they run hot considering casing is finned aluminium.
I don't have an AMR but it is near identical size, capacity and construction to the 1950's MAG blower I managed to get my hands on a while back and have fitted to my vw trike.
No experience of blowers at all so nothing to gauge against as such. Kind of figuring it out as I go along in much the same way as my previous blow through turbo bike build. Being on a small scottish island I don't have many supercharged vehicles around to look at.

Asked a few friends who run supercharged motorcycles down south with small roots style blowers and they have basically said `damn hot after a couple of minutes running`. I'm running in draw through configuration at 2:1 gearing (3.5psi on gauge) and blower casing is out in the breeze. Trike isn't finished and legal yet so can't get out for a longer run to see how hot it runs with proper airflow over it. Just running in the shed for a few minutes at a time as don;t want to overheat the aircooled motor. Gets hot to touch. No way of measuring it but similar to a motorcycle side engine casing after a run. Can put palm of hand on it but uncomfortable after a couple seconds. Inlet manifolds toasty warm too due to hot airflow.
Rebuilt the blower with new bearings, etc. Zero binding and spins freely. Even when hot I can hold the crank pulley in my hands and turn the engine back n forth turning the engine, alternator and blower at the same time between compression strokes. No binding anywhere, belt is loose as toothed cog/belt setup (no slipping obviously).