Yes, unless he has
Raging Vitality (also called the barbarian rage feat tax...), but he has to have enough rage rounds left over to continue to rage or else he loses those temporary hps and can easily die.
Note: These are the Pathfinder Rules not 3.5 FAQ, so if you are playing Pathfinder this is the RAW, if you are playing 3.5 then go with the 3.5 rules.
From
http://www.d20pfsrd.com/classes/core-classes/barbarian under RAGE.
[h=4]Rage (
Ex)[/h] A barbarian can call upon inner reserves of strength and ferocity, granting her additional combat prowess. Starting at 1st level, a barbarian can rage for a number of rounds per day equal to 4 + her
Constitution modifier. At each level after 1st, she can rage for 2 additional rounds. Temporary increases to
Constitution, such as those gained from rage and spells like
bear's endurance, do not increase the total number of rounds that a barbarian can rage per day. A barbarian can enter rage as a free action. The total number of rounds of rage per day is renewed after resting for 8 hours, although these hours do not need to be consecutive.
While in rage, a barbarian gains a +4 morale bonus to her
Strength and
Constitution, as well as a +2 morale bonus on
Will saves. In addition, she takes a –2 penalty to
Armor Class. The increase to
Constitution grants the barbarian 2 hit points per Hit Dice, but these disappear when the rage ends and are not lost first like
temporary hit points. While in rage, a barbarian cannot use any
Charisma-,
Dexterity-, or
Intelligence-based skills (except
Acrobatics,
Fly,
Intimidate, and
Ride) or any ability that requires patience or concentration.
A barbarian can end her rage as a free action and is fatigued after rage for a number of rounds equal to 2 times the number of rounds spent in the rage. A barbarian cannot enter a new rage while fatigued or exhausted but can otherwise enter rage multiple times during a single encounter or combat. If a barbarian falls unconscious, her rage immediately ends, placing her in peril of death.