One thing I've caught onto when people are doing their examples. Marigoldran, I wouldnt expect you to know, as you have stated that you have never built an approval building There are buildings better than stadium. Industrial Sector is age of ascension. Stadium is age of expansion. What about the other techs?
Amusement Park provides +15% approval and +1 for each level, +2 Approval to Adj Approval buildings
MegaResort provides +25% approval and I dont know how much each level, +3 Approval to Adj Approval buildings
some important points:
1) +15% is pretty lame for Amusement park, when its age of expansion tech is +6 approval. The +2 to Adj Buildings is nice. to use it effectively, one would have to have several Stadiums surrounding it. Ok, we have the same decisions to make with power plants, coordination centers, etc - usually though a "central" building offers 'almost' as much bonus as using one of the regular top tech buildings. Its the Adj Bonus that makes it worth it.
2) Entertainment Capital Doesnt sound too interesting, no? BUT, unlike Manufacturing, Trade, Research, and Economic Capitals, it adds +25% approval to ALL planets.
3) Worse issue, MegaResorts - I cant seem to build it, even though I have unlocked all techs. It appears to not be an upgrade from Amusement Park. Bug???? Now lets assume +1 each level, the fact that its +3 per Approval building begins to make it shine.
4) then you have Virtual World, adding +25% morale on ALL planets
so, using 2 Stadiums and the unbuildable MegaReport, with Virtual world and Entertainment Capital (total a +75% bonus), gives you 40 morale. Add a 3rd Stadium in the cluster, gets 11 more base morale, 60 Approval.
60 morale points from 3 buildings, having maxed out the entertainment tree? Not bad, no?
and holy crap, you might have to invest in Benevolent Dignified, only a Tier2 Benevolent Ideology, giving +50% approval to all worlds (with such huge games, I dont see why this would be a terrible sacrifice), which would get you 72 morale points on 3 buildings (which includes the +125% Approval Bonus total, if you have a MegaResort).
Now, imagine adding STARBASES, which at the highest tech, can give you +5 Approval EACH starbase.
Why people are only using entertainment centers and stadiums, just eludes me. Sounds like lots of people havent not alone tried to build approval relief, but have even failed to research it.