Hopefully your 8600M has a temp sensor on it. Grab this, GPU-Z Video card GPU Information Utility
run it and click on the Sensor tab, if there is a temp sensor it should report it. If so put a check in "Log to file", then run your game. Once it starts slowing down check the log and post back what the highest temp was; also what it is/was before running the game (idle).
If it is what I think it is that is happeneing, the only way to fix it will be to contact Asus about an RMA. In case you don't know what I am referring too, http://www.techcrunchit.com/2008/07/...efective-gpus/
and yours is one of those.