Hi there,
 
It sounds like it could be an add-on trying to check for an update when you open the new instance, for this I'd suggest going over all the add-ons (if any) you have installed in firefox and see which one's are set to check for updates at startup, then change them so they don't check for updates at startup maybe.... OR perhaps it's Malware.. unfortunately these days Malware is getting rather good at avoiding detection.. for this I'd suggest running some scans with whatever Internet Security/Anti-Virus app you have.. if you don't have any than I'd suggest using something like NOD32 or one of the free alternatives like AVG or Kaspersky... etc... Hopefully this is helpful in some way... 

 It really sounds like an add-on problem...