No duration was given, so those temporary hit points lasts until she healed back above 0 hit points, or she dies.
Example: Your 10th level pet was struck by an ogre's club and went down to -5 hit points, the Tenacity ability kicks in and it imediatelly gets 10 temporary hit points.
If your pet is healed above 0 hit points, she is no longer dying, as her actual hit points are above 0. And her temporary hit points disapear, as the ability's condition was lost:
a bodyguard with fewer than 0 hit points
Just keep in mind that if your pet happens to go under 0 hit points again, the total temporary hit points will not increase, it remains capped at 10 (the master's level).
All damage should be taken from temporary hit points first.
According to the rules for temporary hit points.
Certain effects give a character temporary hit points. These hit points are in addition to the character's current hit point total and any damage taken by the character is subtracted from these hit points first. Any damage in excess of a character's temporary hit points is applied to his current hit points as normal. If the effect that grants the temporary hit points ends or is dispelled, any remaining temporary hit points go away. The damage they sustained is not transferred to the character's current hit points.
When temporary hit points are lost, they cannot be restored as real hit points can be, even by magic.
When your character is under 0 hit points total, she is Dying until he recieves healing or is stabilized with a Heal DC 15 check. Temporary hit points are not healing, it is simply hit points that should be taken before your actual hit points, as a second health that covers your actual health from taking damage.
So your pet is still dying and losing 1 hit point per round until she recieves at least 1 point of healing, is stabilized, or dies. The temporary hit points are not affected, but will delay her death if he takes any kind of damage, even this hit point loss, it will even prevent her from bleeding to death again if she was made stable and then wounded again.
This loss of hit points work just as regular damage.
This seems to be backed by James Jacobs on this post. But to sum it up:
- They regain consciousness if the temporary hit points would put them at or over an effective hp of zero.
- They do not stabilize, so they continue to lose 1 hp per round until they stabilize.
- They lose 1 hp per round, but they lose hp first from temporary hit points.