
Quoting
L-1
If the lease was in both your names, the obligation to refund half the security deposit to your cousin rests with the landlord. Similarly, the landlord should not have allowed you to move into the new apartment until you paid him whatever balance should have been added to your half of the security deposit for the new place.
While your cousin can sue your landlord for return of his half of the deposit, no doubt the landlord believes he had an understanding that by applying it to your new security deposit, you would refund the amount in question to your cousin on his behalf. You say you don't have the money at the moment. But, I suspect finding it right away and paying your cousin will be much less painful that how much your relationship with your landlord will be damaged if your cousin sues him because you failed to make payment on your landlord's behalf in a timely manner.
As far as receipts are concerned, remind your cousin that you accepted his cash and paid it to the landlord in the form of a check solely as a courtesy to him because he does not maintain a bank account. That doesn't make you his landlord, it simply makes you a friend. If he needs confirmation of rents paid, he needs to seek it from the person he leased the apartment (which coincidentally is the person he needs to sue for his unpaid security deposit). Hmmmm, how do you think that will work out?