The gnome's Alternate Racial trait states
Vivacious: Some gnomes recover 50% more hit points (minimum 1) whenever they recover hit points from rest. Whenever they are healed of hit point damage by a spell, they heal an additional amount equal to 1/2 the spell’s caster level (minimum 0). The extra healing does not apply to spells that grant fast healing or similar effects. This racial trait replaces gnome magic and keen senses. Source PZO9280
Fast healing states
A Creature with the fast Healing special quality regains Hit Points at an exceptional rate, usually 1 or more Hit Points per round, as given in the creature's entry. Except where noted here, fast Healing is just like natural Healing.
Finally, Natural Healing states,
With a full night's rest (8 hours of sleep or more), you recover 1 hit point per character level. Any significant interruption during your rest prevents you from healing that night.
If you undergo complete bed rest for an entire day and night, you recover twice your character level in hit points.
Assuming the Gnome gets fast healing from some source, that is 2 or higher, does this increase apply to fast healing? From my initial reading it can, as far as I can tell, but I have been struggling to find clarification or a straight answer.
Best Answer
Not likely.
Part 1 (Me Complaining)
If we look at Fast Healing, we see as you noted:
At which point the natural inclination is to look at the rules for Natural Healing:
Nothing about fast healing seems to change the need to rest. This means the character needs to be sleeping/resting in order to gain the benefits. Note, that it can be argued that fast healing does change the required timeframe for the rest though, from 8 hours to 1 round. This would mean that a creature would only need to rest for a round in order to gain the healing from fast healing.
Part 2 (Getting To The Point)
Now, it's silly to think that fast healing requires a creature to go ahead and sleep in combat in order to take advantage of it. I've never seen it run that way, nor have I even considered that could be a take before reading these rules today. And while it kind of makes some sense, fast healing speeding up the natural healing process, but still requiring sleep to heal, I don't think that's how I'd recommend running fast healing. Which leads me to my next point, read the ability.
If we look at Vivacious, we see:
See, there's a qualifier there, the ability only applies to hit points gained from resting, not hit points gained from natural healing (which just happens to require rest). While fast healing functions like natural healing, the way I've seen it run, it does not require rest to heal. As such, it's not hit points recovered from rest.
TLDR; (The Finale)
Vivacious does not apply to the healing from fast healing, unless your game makes fast healing require you to be resting for it to apply.