It does sound like you're describing something similar to what that guy in the video with the USB fan was experiencing. The Surface Pro devices are neat for what they are, and they pack a good amount of computing power into a very small package, but gaming can apparently produce a bit too much heat for them to dissipate on their own, resulting in thermal throttling once the heat builds up to a certain level.
It might be possible to hook an external GPU up to that HP ProBook, although its CPU might be a bit weak for some games. I have not personally tried to wire up a desktop GPU to a laptop myself. Typically, I believe they connect to either an ExpressCard slot, or the laptop's mini PCIe slot that would normally be used for its internal wireless card.
Looking up some specs, I believe that laptop should have an ExpressCard slot available. It also sounds like it may use a "PCI Express Half Mini" connection for its Wifi card, though I don't know whether "half mini" would be compatible for this purpose. I did come across a site with a lot of information related to the topic though...
http://forum.notebookreview.com/threads/diy-egpu-experiences.418851/
In general, depending on the connection that the adapter uses and what your laptop supports, you need to either plug it into an ExpressCard slot, or open up the laptop and plug it into the Mini PCIe slot where the Wifi adapter currently is. Then you plug a graphics card into the dock, and a desktop power supply into the graphics card, and then you can connect the GPU to an external monitor or television. Here's a video someone did about it, though it's a bit disorganized and he encounters some compatibility issues between some of the hardware he tested...
https://www.youtube.com/watch?v=jFWnOJSbHiU
You might have some better luck getting answers in that long thread at NotebookReview, since it looks like a lot of people tried that option there.