This is cool. I knew a guy who literally painted gallium onto his processors as thermal paste. He said it worked really well.
They're very effective, but must be used with extreme care because aluminum is highly soluble in liquid gallium. Aluminum is the most common material for heatsinks, and it will literally dissolve if the gallium touches it. Gallium is also electrically conductive, so if you accidentally dripped any into your processor socket I assume you're going to have a problem.
If you're careful with it and make sure to get a cooler with a copper contact area, it'll cool more effectively than traditional thermal pastes. I thought about trying it, but it seemed like more hassle than it was worth.
Aluminium is the most common material for heatsink fins. The cold plate is almost always (nickel-plated) copper.
https://www.youtube.com/watch?v=4HKpMYJ-6go
Did he apply it directly to the top of the chip die or to one of those aluminum package covers? Did he use a copper heat-sink? Is somebody's leg being pulled?
The other side (heatsink with fan) probably had to have been full copper (not the pipe/aluminum)
Either way, it doesn't seem worth the effort for marginal gains.