Check your lease agreement to be sure, but generally this is the responsibility of the owner/operator to fix. They may want you to call their maintenance man yourself, but they should pay for it.
In most states it is the responsibility of the landlord to ensure that the apartment is habitable. This should include most major mechanicals, including plumbing, electrical, and a/c.
The apartment management should be the ones responsible for repairing any wear and tear that the apartment may suffer. I recently had a hole in my tub patched up and the apartment complex was the one that paid for it to be performed. I hope this answer was helpful to you.
In almost all rental situations, mechanical breakdowns such as this are the responsibility of the apartment manager or owner. You may want to double check your lease to be sure but the tenant should not be liable for damage to Furnaces or air conditioners.
