Mike,
I have not heard what B&C's response to your problem was, but I had a
similar problem with my 360's B&C starter after about 150hrs. However,
it was an intermittent problem, NOT particularly related to hot
conditions.

When I called B&C, they told me that they had had some problems with
light corrosion on the solenoid plunger/barrel.  They suggested that I
remove the solenoid and clean the plunger AND the barrel.  I used
acetone and a Scotchbrite pad (did not notice much/any visible
corrosion).  They then suggested lubing the plunger/barrel with a
silicone grease (Dow DC4 if memory serves correct).  

Since following their instructions, I have flown 200hours trouble free.
Note that it is a little tricky to reconnect the solenoid lever arm.
Regards,
Clark Baker
N250JF, LNC2 360
450hrs