Proof of Stake - double spending
What you are describing is a well documented problem with Proof of Stake called Nothing at Stake theory. 


  In PoW they could check the hash, but they can't here.


That's incorrect. You could produce a valid hash and still have double spend transactions in PoW. That is because bitcoin blockchain is designed to accept the longest chain (technically, chain with the most proof of work). However, to launch such an attack in a PoW chain, the attacker must either control >50% of hashing power to mine faster than the rest of the network or split his computing power in half if he wants to mine on competing chains. That is not economically sound decision and hence cannot be achieved with PoW. However, in PoS the shortcoming arises because it costs a validator nothing to validate transactions on multiple forks. 


  How to prevent Nothing at Stake Attack?


There have been proposed theories that involves penalizing validators if they simultaneously create blocks on multiple chains, by means of including proof of misbehavior. This means malicious validator's deposit is deducted if two conflicting signed block headers are found into the blockchain as a later point in time. This can be achieved because, to qualify as a validator, nodes will have to lock certain in the chain through a smart contract.