This cabinet uses a wall cleat. If that cleat is secure then the cabinet will be secure. If it is not secure then the whole thing will come crashing down when you least expect it.
Two options:
One Stud
Drill a hole in the cleat lined up with a stud. The catch is you have very little height to work with, as you can't put a screw through the angled part (because then the cabinet won't fit on it properly). Which means you can really only put in one screw. Use a nice big cabinet/construction screw, not a drywall screw. (Drywall screws are for attaching drywall to studs. They work perfectly fine for plenty of other light-duty things but they are far from ideal for hanging cabinets, especially when you can only use one screw.)
The screw should be long enough to get through the cleat, the 1/2" (likely) of drywall and at least an inch into the stud. Which means probably a 2-1/2" or 3" screw.
Then use anchors for the other positions just to keep it from rotating, though snap toggles or similar would be much better than simple drywall anchors.
Add Backing Wood
Mark an inch or so in from the left and right sides, all around, where the cleat will be mounted. Cut a rectangle of drywall out. For ease of access, probably cut down several inches below, but not above because the cabinet hangs at the very top.
Now you have access to the wall. Add additional wood (2x4 is perfect but you can use something a bit smaller to make it easier to get it through the hole and mounted) to span between studs at the height of the cleat so that you can put screws through all three holes of the cleat. Put the drywall back in. Patch/paint? I wouldn't bother, but if you think this is "temporary" and might be exposed in the future then do at least a basic drywall patch job.
Install the cleat with screws into the new backing wood.