Why does a sugar substitute that dissolves when a drink is hot stay at the bottom of the cup when it cools down?

The higher the temperature of a liquid, the more of a substance it can dissolve. So when a liquid cools back down, often some of the dissolved particles re-form into solid and precipitate out of the solution. This applies to sugar substitute as it does to any other substance. Just like regular sugar dissolves in your coffee better when your coffee is hot.