There are log files that refresh every time the game is closed. Immediately after that happens again please close Demigod and follow this link in Internet Explorer to gather those log files:
http://sd.stardock.com/Demigod_report.zip
Save the file, run it, and attach the .zip file (located under YOUR USERNAME\My Documents\My Games\Gas Powered Games\Demigod\report) it generates to an e-mail to support@stardock.com for further support.
Some email services restrict .zip attachments. If your email won't allow you to send the file, simply change the file extension from .zip to .zzz, and it should send without issues.