For me it seems planets always grow 3 health per time unit, with TEC research level2 4 (sometimes 3), it never increases nor decreases in any other ways. No matter if they have 1000000000000 max health or just 500. These are just my observations from the game, not from any files or something.
While being bombed it never increases, and if bombing cedes, it takes several seconds to reinitiate health growth. So it is useless upgrading the planet IF YOU CANNOT DEFEND IT, in other words you cannot make your opponent leave the planet alone (scare him away or destroy sieging ships), or a Novalith warhead is coming.
BUT if you can stop his siege, and make him go away, and bombardment will not be likely for a longer time as you just made a minor or bigger victory, it worths it if it can go above the original max level before the next round of sieging. But it seems situational, better to upgrade in advance and let it grow to a high value before anyone can siege it, of course if it will be a frontline world that your enemy will surely want to take.
For me it's some kind of rule, never upgrade planet population if a serious amount bombardment is imminent, you just lose those resources if you lose the planet. But this goes for only AI, online I never really cared about planet health ever.
EDIT: or does it grow always 4 units and with TEC research 5? Ah I can't remember correctly.
There is actually a per-planet-type health ("emergency facilities") growth rate in the entity files. It's called healthRegenRate and is set to 6.0 for all colonizable planets (although in theory it could be different for each type.) Given what we know about rates in general in this game, it's fairly safe to assume that figure is per second (rather than some other unit of time.) I've actually timed how long it takes for 500 hp to grow on a roid after colonization [making sure it doesn't get hit by any bombs, so I actually timed how long it took from 500 to 1000 hp] and it took 83 seconds... so no surprise there, the rate is per second. The game probably updates the displayed planet health every half-second which is why it seems to grow in chunks of 3.
In the entity files, healthRegenRate is a constant however for each planet type, i.e. it can't be set depending on the max-health of the planet or its development level of some kind (not in straightforward way anyway.) There's only one tech that increases that rate, TEC I think, and it grants 12.5% bonus per research level (max 2) via PlanetHealthRegenRateAdjustment. There doesn't seem to be any [random] planet bonus or artifact affecting that rate.
So, for optimal growth of planet hp in front-line situation, one should begin the emergency facilities when the planet is 270 hp below its present max; that's at the first level of upgrade which takes 45 seconds to complete (270=45*6). Subsequent levels add 15 seconds to upgrade time, so one should add 90 hp per level to the gap from max before upgrade should begin, so 360 hp away from the max is optimal for the 2nd upgrade and so forth. The (rarely researched, I assume) TEC boosters increase the hp growth to the absolute rates of 6.75 and 7.5 respectively, so the optimal point for the first level moves to 303.75 (below max) and respectively to 337.5 for the 2nd level of said research; the extra hp to subtract for the higher levels of hp upgrade also goes from 90 to 101.25 and receptively to 112.5 with these researches, so around 400 hp below max would be optimal for the 2nd upgrade with the 1st research booster and 450 with the 2nd one.
More importantly, on the "faster" ship speed setting, one should not bother beginning to upgrade emergency facilities once an enemy cap has entered the well (and is headed for the planet), because it takes just about 45 seconds for the slowest kind of cap (like a carrier) to close the distance to the planet enough to begin bombarding it. And since planet hp regen is disabled during bombardment (and also because the planet will typically be well below max hp by the time the enemy cap is chased away if that even succeeds), that's money better put to other uses right away... But if you can spot the enemy cap in the next well coming your way (by scouting!) then it normally pays off to begin upgrading the max hp ("emergency facilities") if you haven't done so already. The time for the typical roid to get from 1000 to 2500 hp, thus to maximize the benefit/payoff of the upgrade before bombardment begins is however a substantial 250 seconds (4 minutes and 10 seconds)... so it is better to have significantly more advance notice of the enemy's arrival when starting the upgrade. Typically a cap ship takes about one to one-and-a-half minute per well to traverse, including the jump phase, on "faster". So notice while the enemy is still 3-4 jumps away is needed to maximize payoff/hp from [the] one [and only] level of roid hp upgrade if the enemy is heading toward your roid without pausing.