The external amplifier will help reduce the load on the Marantz SR 7013. But the best way to reduce heat is with these fans:
Amazon.com: AC Infinity MULTIFAN S7, Quiet Dual 120mm USB Fan, UL-Certified for Receiver DVR Playstation Xbox Computer Cabinet Cooling: Computers & Accessories
www.amazon.com
In most cases, you will not notice much difference with the external amp(s) except at
extreme volume levels. I have the Denon 4700 and use it in preamp mode with a 5 channel (Adcom) and a 7 channel (Monolith X7). While I went this route, it adds quite a bit of expense and increases energy usage for not as much difference in sound as you might think.
I would suggest getting the fans and try using your speakers on the AVR. It may work well for you? The other direction of adding external amps and maybe even moving all speakers off the AVR is expensive and most likely will not provide as large a difference as you are expecting?
It's a journey that many have tried....