Does not work anymore.
Glad you raised the cross-device behavior issue. IMHO the double tap should be active only on mobile, otherwise hovering is the way to display and click the way to go, don't you think?
I know this is a little old, but I recently came across this while looking for a similar solution. Relying on hover events isn't good for responsive design, and especially terrible on mobile/touch screens. I ended up making a small edit to the dropdown.js file the allows you to click the menu item to open the menu and if you click the menu item again it will follow it.
It is likely that bootstrap's own dropdown.js has diverged enough in the 2+ years that this version no longer applies. The version linked above was for bootstrap v3.3.5. It looks like the eventing in 3.3.6 has changed a bit, I haven't tested against that version. Also, it surely won't work in 4.x.
The nice thing about this is it doesn't rely on hover at all and so it still works really nicely on a touch screen.
This is by far the best way to go. Thanks for doing this, I was just about to do the same thing, you saved me from re-inventing the wheel!
Usually you'd have scripts that will disable hover functions on mobile sized devices, so having hover really isnt such a bad idea, just got to make sure to alter the hover once the resolutions are mobile sized.