I am running a campaign where my characters need to destroy a roughly cylindrical stone, about a foot in diameter and a couple feet tall, with a hardness of 7 and 28HP (14 before it is broken). Hardness seems to act like resistance in general, but I would have thought that stone would have even greater resistance to slashing or piercing damage, than to bludgeoning damage. Is there any support for this in the rules, or has anyone just done it anyway?
The hardness rules appear on page 272 of the CRB. They do indeed work against every type of damage equally, as you suppose: “Each time an item takes damage, reduce any damage the item takes by its Hardness. The rest of the damage reduces the item’s Hit Points.” Page 273 adds to that Object Immunities, but I don’t think they apply to your question.
If you want to homebrew this differently, please keep in mind that this affects shields as well.