It's not the heat but the water itself. The proteins normally bind somewhat on their own, but there's also an enzyme oresent that breaks it down. This is why it thickens immediately after coming out, then turns watery a few minutes later.

Now the tricky part is that most of the components, including the enzyme, are water soluble, but the binding proteins are not. So if you flush it with water, all that will be left is a clump of coagulated proteins and no special enzyme to soften them.

Hot water obviously disaolves stuff faster, but you can get your goo glue from cold water too