| do multiple entertainment areas stack. |
Yes.
BUT (and it's a big but) each entertainment improvement's effect is diminished as population grows. Now, I know that the formula for planetary approval that is posted in the wiki is badly flawed and outdated, but it still contains a grain of truth.
Here is how I believe it works, based on what I have read and my own experiments: There is a negative approval modifier for population. It goes as low as 20% when your planetary population hits 25. This modifier affects not only the planet's
base approval rating, but also the effectiveness of
each entertainment building. So, at 25+ population, that big, fat VR Center that says it gives 60% approval is really only adding 12% (20% of the full value) because of the population modifier.
If anybody who reads this knows better, please enlighten us. The whole twisted, incestuous relationship between population, approval, and planetary improvements is even harder to untangle than the Reaganomics that govern production, research, and income.