@Dave Fritz that appears to be a diaphragm pump. I wouldn't expect the diaphragm to get hot, and you say it's the motor that is getting hot. It does look like maybe one of the warning triangles in the sticker on the motor case might be a "this gets hot! don't touch" kind of warning, in which case maybe it is behaving normally.
Still might be worth checking to see if there is a cooling fan at the outboard end of the motor and if so that all is well there.
Underneath the disk thing to which the hoses connect (i.e. the diaphragm assembly), there is a silver "box" at the other end of the motor which probably contains some kind of "gear box" or ellipitcal drive that moves a shaft up and down which in turn actuates the diaphragm. You might look for a port or two (upper and lower ones?) that might be for filling that "gear box" with oil.
You could also try peeling the Stubby Lathes sticker off to see if the information hidden underneath it might provide some further insight.
Good Luck.