There's one bug we're working on, but it's rare and doesn't cause any crashes. If you're using the latest version, crash bugs are very, very rare in Hegemony. The most common cause of the game crashing 5-10 minutes in is a video card overheating.
The first thing you should do is update your video drivers.
After that, there's different tools to check your video card temperature.
If you have an Nvidia card, install the Nvidia System Tools, then right-click on your desktop and choose "NVIDIA Control Panel" and choose the "Performance" tab. If you haven't already accepted the license agreement, do that.
If you have an ATI card, right-click on your desktop and pick 'Catalyst Control Centre'. If you're not already in advanced mode, click the Advanced button. The in the top-left corner, click the drop-down and select 'ATI Overdrive'. Click the key button to enable these features.
In both cases, you should now be able to see the temperature of your graphics card. Play the game until it crashes and then check the temperature. If it's below 70°C, that's fine, and if it's above 85°C, that's a serious problem. Anything in-between is a bit worrying, but depending on your computer, not necessarily a problem. You can also change the fan speed, and increasing it might help. Note that, for some reason, the Nvidia Control Panel seems to require that you set the fan speed twice in a row for it to take effect. (You can usually hear the difference when the fan speed changes.)
It could also be possible that your CPU is overheating, but nowadays a graphics card overheating is far more common.
(Note: The latest version of Nvidia System Tools at the time of writing (version 6.06) appears to have a bug where it may not show the temperature properly. If this is the case for you, I suggest using GPUZ to see the temperature.)