A metal aerogel would have very high surface area (much like adsorption humidity control crystals) but that's not really helpful when you then need to move the fluid touching all that surface area in order to remove the heat.
Typically the fluid will be air which sucks at moving heat (hence why aerogels are great insulators as they hold a lot of air).
I'm not positive but I suspect that an aerogel in a higher specific heat fluid wouldn't do any better as the flow volume through the aerogel would be quite low bounded by the volume of the aerogel.
If you want to cool the hotside of a TEC you need to push a lot of watts per unit area. The best things for that are high specific heat fluids which are often used in compressor systems and then they're termed refrigerants so you're back to traditional air conditioning. The other option is very cold fluids (liquid nitrogen) but at that point you may be better off bringing the LN2 into more direct contact with whatever you're trying to cool.
An aerogel would be among the worst things to use as a heatsink. They are mostly air, held in place. Air has a very low thermal conductivity, low density, and low heat capacity. Air cooling is used because air is abundant and usually readily available, and relies on convection, which an aerogel prevents.
A much better solution are heat pipes, which utilize convection internally to create very high effective thermal conductivity. Assuming the system is not in continual use, the need for a large radiator can be avoided, by using a tank of water or oil (if electrical conductivity is a concern) to dump heat into. With a tank of water, you can also cool below ambient temperature by using evaporative cooling.
Typically the fluid will be air which sucks at moving heat (hence why aerogels are great insulators as they hold a lot of air).
I'm not positive but I suspect that an aerogel in a higher specific heat fluid wouldn't do any better as the flow volume through the aerogel would be quite low bounded by the volume of the aerogel.
If you want to cool the hotside of a TEC you need to push a lot of watts per unit area. The best things for that are high specific heat fluids which are often used in compressor systems and then they're termed refrigerants so you're back to traditional air conditioning. The other option is very cold fluids (liquid nitrogen) but at that point you may be better off bringing the LN2 into more direct contact with whatever you're trying to cool.