When you look at all my calculations, a lot hinges on average influence per influence generating planet. You can see the difference when I assume it to be sqrt(30) vs sqrt(36) or just 8.
Also, the more of your IP earnings comes from population, not flat bonuses, the worse you are in terms of colonizing new planets (another thing that makes flat bonus like from artisan workshops very nice). It is likely an intended feature, however it puts relatively more pressure on early game, making it look worse than idealized from some end state as shown in my charts.
Putting sqrt(x) actually makes the early game even harder than just "x" - partly an intended feature, to bring soft cap sooner, but it also means comparatively lower percentage of productive planets at the beginning compared to linear decline in case of just "x"
The harder the game is at the start, the bigger part starting luck plays. At the same time it puts even higher emphasis on knowing what you are doing.
However, the part about knowing what you are doing kind of stays with you all the time, while the part about better starting luck is by definition only at the start. So generally I support sentiment of easing out on the early game. Something like not paying influence for your capital, and having a flat influence bonus for it instead, is one thing to ease out the very start, for example.
Looking in terms of influence. The formula for cost per colony being at the center of considerations, you may put like this:
P * f(x) * g(y)
where P is some scaling factor, f(x) is some function on number of planets you have x, g(y) is some function of your influence y. Oberlus proposal adjusts f(x) to galaxy size. The function being actually a function on percentage of galaxy owned, rather than on number of planets, makes a perfect sense to me. The other function, g(y), is put in terms of average influence per average planet, which also makes sense to me. Although, the proposal makes it to be a function on "how much an end game civilization is expected to produce on average planet", "Max", and that's like 8 or 9, apparently.
Now P is something left to gameplay settings. Oberlus understands the parameter as percentage of galaxy an end game civ should be able to grab at hard cap (hard cap is when all planets need influence focus) - assuming by definition it is able to do the 8 or 9 average influence per planet. So if you set it to like 75% then it would leave 25% planets left on "productive" duty when colonizing entire galaxy, while something like 200% means that hard cap is on 50% of the galaxy so you can't really colonize everything.
But the equation can also be viewed as putting that percentage on g(y). So a 75% setting is what allows an empire that does 75% of "Max" to colonize 100% of the galaxy.
Lower P makes early game easier, it actually makes entire game easier.
Daybreak, I understand that you are proposing playing with the g(y) part. However, what you are proposing seems dangerously close to g(y) meaning "average influence per planet you are able to muster at the given time". So yes, it is possible to factor into the equation, but it actually factors out of the equation entirely the "average influence per planet" part. Not only if your start was good or not, but also if you bother in any way to have your influence game improved or not. This is concern expressed by Oberlus
Now, we are not 100% sure what "Max" is, so while g(y) remains a constant, it is left to players to decide (though if it is a constant, then changing P appropriately can have the same effect as adjusting "Max").
I am not sure if there is a way to make g(y) dynamic. If you tie it to influence output, then it would seem like a punishment on people who actually improve their influence game. On the other hand, f(x), it being a punishment on number of colonies you have, is actually the intended feature.
Now, the aim is to improve early game, right? So why not introduce a time component in the cost function? Well, maybe. But then you'd get an effect where empire that does not expand, or does not improve their influence game in other way, eventually crumbles. Well, maybe that would be an interesting feature, but don't know. Of course this could be something adjusted to some max number of turns, now that sounds better. This would need to tie generally to some "game pace" setting in general. But I don't have any concrete proposals at this time, just an idea.
But playing with actual time component seems too tricky to me. There are already implicit things about it - research takes time, colonization takes time... Population growth takes time... Function on number of owned planets f(x) has this baked in a bit, with the assumption that number of planets relates to time, but then if you lose planets... you don't actually go back in time, do you? So would the time component still make sense anyway? To distinguish between small empires and early empires directly?
Influence part has some baked in time component, too, but in terms of research, unlocking polices, finding interesting planets etc. Maybe we should have some relatively late game but cheap research option to improve influence/lower cost per planet, flat, to better distninguish between early empires and late but small empires? But then you'd adjust your Max accordingly, making early game even harder, then you'd need to perhaps still improve early game in other way. And such boost would boost large empires as well... So need to figure out how to boost smaller empires comparatively better (flat boost? some nonlinear function actually better for smaller empires?)
In the end, I would first go thinking in terms of shape of the curve of usable planets. That's why I was showing the graphs earlier etc. sqrt(%planets) seems actually punishing for the early game to me, because smaller empires have comparatively lower percent of usable planets, not only because soft cap is hit sooner. So initial expansion is actually punished relatively more, up to the sweet spot, the soft cap, where any expansion is punished just in general in terms of usable planets. Maybe that's the graph I should be showing. Conversely, squaring the % of galaxy owned makes you have comparatively more usable planets in the early game than late game, but it also moves the soft cap. But we could be using a curve that rewards initial expansion more/gives relative boost for smaller empire in % of usable planets (I appreciate such curve could also be some kind of policy that shifts "wide" towards "tall", giving even better % to small empires).
I could give a think about a better curve, if there is interest? I may have a think even if there is no interest
Something with inflection point... So while percent of usable planets drops as you acquire new planets, it drops slower at start.
I gave some initial think at least to show what I am talking about, below is number of usable planets for galaxy size 200 with sqrt of percentage of galaxy owned, just percentage of galaxy owned, and my "secret function" that is along what I'm thinking. Like the "secret function" on this example soft caps even faster, but at higher number of planets than sqrt (lower than "just x"), and at first it has better ratio of productive planets to owned planets than sqrt or "plain x"