That’s a great question. A double top is generally an indicator of strong resistance. As with all resistance indicators, I believe positions should be monitored a bit more closely near these areas. At this strong resistance level it is generally likely that the stock will consolidate, then fall back downward as it did when it hit this resistance level previously. Think of it as if the stock is underneath the ice of a frozen pond. As it swims along, it may try to break through the ice but often has a hard time doing so and therefore dips back down, possibly attempting to do so again at a different spot (in time).
Keep in mind, however, that just because the stock has performed a certain way historically, a repeat of history may be probable but it is not at all guaranteed. What I mean is, if the stock is able to break above a strong resistance level (due to increased investor confidence, etc.), this is generally a significant event. Again with the frozen pond analogy; if the stock is able to break through the ice it is likely to then use the layer of ice as support to stand on. In other words, if a stock does break above resistance it is likely to use the former resistance level as support, moving forward.
As mentioned, resistance levels, in my opinion, are very significant. They can serve as a good place to sell an equity or, conversely, if the stock breaks above resistance, this can be a good place to buy/hold. Generally, with the case of a resistance ceiling (in the form of a double top, or otherwise), I like to place a tight trailing stop on the equity. What this does is, if the stock does what it has done historically and falls after hitting resistance, my trailing stop is executed and I am out of the position. Or, conversely, if the stock is able to break above the resistance, my trailing stop is not executed and I stay in the winning position.
Hope this helps clarify the significance of resistance and double tops. (All this info also applies to support, with, of course, inverse results.) Here's a video on how to use trailing stop orders around resistance level: