The Rage's duration is 4+Con mod rounds per day. Those rounds shouldnt be continuants, right? And the Con mod is before enter rage. The normal one. Without calculating spells active on barbarian.
But some Power Rage abilities use the Con modifier. Which one? The new (after the rage) or the normal one? Various spells like bear's endurance add to this con mod?
For example the Renewed Vigor. Which Con mod the barbarian should use to heal HPs?
All rage abilities should use the character's new modified rage unless specified by the text of the ability. The actual Rage ability as stated in the text does not use the temporary increased constitution score. A first level barbarian with an 18 constitution can rage for 8 rounds, they can be continuous rounds or he can divide them up into 8 seperate instances of rage, 1 round each. At second level this would increase to 10 rounds.
Renewed Vigor should use the barbarian's increased constitution modifier.