Currently, EEPROM support seems to be limited to the currently active session, that is unlike an actual EEPROM, gpsim core don't seem to support persistence for data stored in memory regions designated as "EEPROM".
I think it would add a good amount of realism, if gpsim could optionally support inter-instance persistence for EEPROM data.
Something like this could probably be easily achieved by simply serializing the corresponding bytes to a simple ASCII file (i.e. simply write the hex values as memory/data pairs: 0x00:0xFF, 0x01:0xCD, 0x02:0X02 etc.).
This ASCII file could then -for example- be referenced in the assembly file by using a corresponding pre-processor directive, or alternatively simply by providing a corresponding new switch to gpsim during startup.
Of course, different PICs have different requirements, so saving some additional meta information, might also be a good idea (i.e. type of PIC, so that the valid memory ranges/addresses can be derived, and possibly show a warning if an EEPROM data file contains value pairs for memory locations that aren't valid for that particular PIC).