As long as Horizon Stone remains under your control, you'll retain unspent mana as steps and phases end, although that mana will become colorless. This means you can add mana and spend it during a future step, phase, or turn. Once Horizon Stone leaves your control, you'll have until the end of the current step or phase to spend the mana before it is lost.

If unspent mana you have has any restrictions or riders associated with it (for example, if it was produced by Jeweled Lotus), those restrictions or riders remain associated with that mana when it becomes colorless.