AP vote intervals aren't random. Generally the interval is twice as long as the UN's interval, so usually around 10 turns.
However, unlike the UN there's the chance that you can be the only AP member (since its based on religion instead of just every civ), or that none of the possible votes are applicable (again based on religion percentages). If either of these are true, then naturally there's nothing possible to vote on, so it skips that interval.
Spread the AP religion to more civs / cities.