I always thought a lot, like twice as much as iron/steel if not more. However I can't find anything (in the SRD anyway) saying how much so does anyone know?
Since objects made of adamantine don't change their weight, yeah, it probably should be around the same weight as steel.
On the other hand, it could be heavier and the added weight per volume could be compensated by adamantine not needing as much material to get the same effect - maybe adamantine is twice as heavy as normal steel, but adamantine armor and weapons are only half as thick as regular objects, but still work because adamantine is just so durable.
But it's probably simply just as heavy as regular steel, it's easier to handle that way, and I don't think it's ever specified.
Pathfinder is an exceptions-based system, so if the rules don't specify a weight change, it stays the same as a normal item. Mithral and darkwood change the item's weight; adamantine and alchemical silver do not.
This is confirmed by the magic items chapter, where the adamantine battleaxe, dagger, breastplate and (dwarven) plate weigh the same as standard steel items.