hot rocking

Anyone else here think that Lavabending in LoK don’t really make sense?

In ATLA, Katara and Toph both used their Water and Earth bending to stop or bend the mud, that it was earth-water material.

Lava is hot liquid rock, means fire-earth material.

So, it requested to bend this material both Earth and Fire benders together. Yet we found Earthbender can heated and cool the lava without any help of Firebender! How is that possible?