I would like to 2nd the suggestion of using thin wood, gluing it to the inside of the door and then gluing another piece over that so that the new piece is level with the door and mostly fills the hole. You then use wood filler to fill in the spaces and make it level. Sand repair and a bit around the repair and paint. You should be all good at that point.
I would also like to recommend that you take the door off the hinges and paint the entire door as the damage will be less noticable at that point.
Also - if this is an apartment you may find that they have used a generic door and that home depot has a door replacement with the proper cuts for the hinges and nob already. This still would need to be painted and mounted so it may be more work than is neccessary.
As a second suggestion the expanding foam one was very interesting - I have used that a ton and it should work easily and provide a reliable solution.
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