Divinity is a reflection of the strength of your godhood. Higher divinity levels will strengthen your command and control of your authorities, allow you to use them more often, etc. I think I'm going to delineate from campione canon soon and allow for the expansion of authorities as campione and gods grow in strength. An example would be if you have very high divinity your authority might evolve or split into two authorities. Still on the fence about this one but considering it.
I think the best way to handle this would be that rather than having an Authority evolve or split the user would either have to put training time in to fully master the Authority and/or reach a sufficient level of divinity to do so automatically. The end result would be that restrictions on Authorities might be lifted, rather than the Authority outright evolving.
1. As a canon example of this Godou is able to banish the Boar after summoning it, but if it hasn't fulfilled its purpose when he makes the attempt there's a chance it'll fight him - this apparently indicates he hasn't fully mastered the use of that Authority.
2. As an in quest example of this higher divinity would let us maintain more high level artifacts from the Golden Pagoda at a given time. Currently as a 'baby' Campione with relatively low divinity we're restricted to one, but raising our divinity will let us take another one out.
3. Various restrictions like "godspeed is physically taxing without using a related Avatar" (a restriction for both Black Prince Alec and Godou), "this Authority may only be used once per day", "using this Authority in this manner will lock it out for half a day" might be lifted or lessened, making them more useful. The main reason for most of the restrictions is that Campione are still somewhat mortal, so making them more divine would logically make them able to use Authorities better.
Another possible way to handle this would be giving Madara the ability to combine Authorities. Godou is the only canon Campione who can do this, likely due to his main Authorities all actually being one big one, though it's taxing. With sufficient Divinity and training Madara may be able to figure out how to do this - his Sharingan may help him understand Authorities in a way other Campione can't, as an in quest explanation.
Of course, for Authorities that lack restrictions you could simply power them up - more divinity -> more divine power -> more powerful Authority. For instance, as I said before with Black Prince Alec's Black Lightning Authority, when he's in the Avatar of Lightning form his Lightning isn't strong enough to damage Campione or Heretic Gods. Higher divinity could up the power so that it could start doing low level damage.