There are technologies that give percentage bonuses that multiply what your stadium supplies. There are approval Relics that supply multipliers. There are ideologies that provide bonuses or buildings that affect approval.
If you are going for that big an empire, you are going to need all of those. If I get a bunch of those then I find rush buying a single entertainment center on a new colony makes a big difference for that new colony.
I have researched all of those approval technologies, including Precursor Relic research tree. I also got nearly all of the Approval relics found on that map, about 10 of them, and they are all being mined to full. Also I have several ideologies that gives me percentage multiplier boost.
And I'm sorry but no, rush buying a single entertainment center does not help and does not make a big difference at all. Not when you have nearly 250+ colonies and you START a new colony with -55 approval points(only from LEP alone). Add all of those percentage multiplier bonuses, still your colonies with anything less than Class 10, are never going to reach a decent approval rate. And like I mentioned earlier, with all the bonuses and multipliers (and bugged multipliers), I get final -95 approval points when I start a new colony (I need to check but its somewhere around -80~ -98 pts depending on colony, it varies).
So far the only solution is the Patriotic trait which removes the LEP completely.
Some users say the LEP is fair, and I can see their point is valid. But then I also see the side where I fundamentally wonder, "Why having a large empire something bad? Why do we have to be punished by having a large empire? Why having a large empire automatically means your citizen are going to be unhappy?"
In my opinion, the assumption that 'large empire = bad = citizens will dislike it' is flawed.
Citizens don't necessarily have to be patriotic to approve a large empire. Even if you're not patriotic, you may like the idea that the empire you live in is big (for your own trade, your own profit, your own shady business, whatever the reason).
I don't think players should be 'forced' to pick patriotic skill if they want to have a large empire, because, your empire is most likely going to become very large at the later game.
Like someone else in the forum mentioned, Large Empire Penalty and patriotic skill was there in place to prevent a problem of accelerated, rushed expansionism. And justly, these two system does not prevent it. Even with zero approval on every colony, you can expand rapidly, colonize as much as you can, and pump out the manufacturing/research/wealth benefits. Therefore, LEP does not hinder or stop the rushed expansionist strategy. If that's not the objective of LEP, then I don't really know what is it there for. Currently the LEP is a bit too excessive in my opinion, and has a harmful effect of preventing a player with large empire to play 'peace/diplomacy/trade/approval' strategy.
Looks like Stardock is assuming that "If you have large empire = you're warlike, you invade a lot, and you're evil empire", which is not always the case. I'm going for a 'large influence/culture-based empire', and this is not entirely possible with LEP (unless i restart game with Patriotic).