I've noticed that IC saves the contents of the "Default" anonymous Apex tab so that the contents of that tab remain available even after restarting IDEA.
It would be nice if the state of ALL other anonymous Apex and SOQL query tabs was also somehow saved, just like the "Default" tab is.
I know that we can save the contents of a tab to disk, and then reopen, so this is by no means a critical request. Just a small "nice to have". :-)