Does Cox Internet stay up as long as Cox Phone during a power outage?

I've been thinking of switching from Cox phone to another voip service like OOMA or Obi with Google Voice.  But in order for OOMA or Obi to work during a power outage the Cox internet signal has to be there.  Even if my cable modem and router and OOMA or Obi box is plugged into a UPS backup power supply, the Cox internet signal has to still be active.  From what I've read, there are regulations that require phone companies to have backup power for their phone system if there is a power outage.  

Does the Cox phone modem lock on to a different signal from the regular internet signal?  And does that phone only signal have a Cox backup power system while the regular Cox internet signal does not?  Or, does the Cox backup power system keep the regular internet up as long as phone?  If I switch to an internet phone service like OOMA or Obi with google voice would my phones work as long as they would as if I were still on Cox phone? 

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