The time of the doubling cube entering the backgammon board is not clear, some say 1920, some say earlier. Whatever the date is, it doesn't really matter. What matters is that the doubling cube changed backgammon, and continues to influence the game today.
Where and when to use the doubling cube that is the real question? The doubling cube is a weapon that when used skillfully usually defeats the opposition. The decision on how to play it though, can lead to the overall outcome of the game or match. The best possible choice is at the top of the market, invariably this is rejected, which leaves either an early roll, or a late roll depending upon the state of the actual game.
One tip is nearly always double when you are near to the top of the market, or beyond the opposition's drop point.
When contemplating the doubling cube, think of the consequences; look at your position in respect of 'could you win without it? The reason for this self assessment is that the doubling cube can come with a cost. If you double early, the opposition has a good opportunity to use it against you. A late double and you could lose any possible advantage that the double makes.
The doubling cube has certainly changed backgammon, bringing a new element of uncertainty and possibility into a game that up until then relied mainly on luck.