Hi!
Move the mouse cursor over the approval rate number in planet window, and you'll see how it is calculated.

The easiest thing to keep your appoval high is to keep pop on a planet low, or to keep taxes low. However you'll get only a low amount of money as a result, as they both "generate" it.

As a rule of thumb I leave pop on smaller planets at its first treshold: 5B. If the planet can build a farm AND a morale building, I consider building them to get more pop (10B or 11B) for taxing. On bigger planets (12+) I go for 15-17B treshold, but usually I need to build there 1.5 morale building per farm. Mining a morale resource helps too - it is worth about one morale building per planet.

If all that fails, I lower the tax rate. Nothe there is a distinctive treshold, when going with tax rate form 49% to 50%. So if you're running them just about 50% and the approval is low, consider droppong them to 49%. HTH.
BR, Iztok

Is there any documentation on the thresholds of where population nos. will start to decrease approval ratings further down? Or is it simply a linear equation?