It would require the silicon to run at higher temperatures than when air-cooled. Semiconductor failure rate roughly doubles every 10C temperature rise. For most environments, it's better to spend energy to actively cool the silicon than extract heat energy from it, because the cost of failures is higher than the cost of power.
I don't know if it's a frequency illusion or if a lot of people are playing with them lately, but I've seen tons of people mentioning peltier cooling lately.
I've been working on a design to cool my cpu with one. The best part about peltier modules is also the worst part: the cold side gets really cold. So cold you get below the humidity threshold, and end up with water all over your system. I'm trying to hook up a bunch of humidity, and temperature sensors to an arduino that will regulate the power to the peltier module to try to only let it get as cool as the relative humidity will safely allow.
What I'm getting at is you could do it, but a peltier cooler would need to be manufactured for it to be reasonable at the scale they're looking for.
I remember them being used for "extreme" overclocking.
Nowadays people in that department seem to have moved to the elegant solution of manually pouring LN2 on to a cup that sits on top of the CPU (or GPU).