What matters is what your lease says and what your local laws say.
It's likely that your lease says that your landlord will provide certain appliances (usually listed) in working condition. And it's likely that the lease says that the landlord is responsible for maintenance and repairs on them. If so, then that establishes that it's the landlord's responsibility.
However, the lease might not say that. It might make you responsible. Or it might say that you're required to pay the first $xx of any repairs. So, the answer really lies in your lease.
How long do they have to make the repairs? Maybe your lease specifies that, but more likely it doesn't. It would probably fall under the same provisions that require your landlord to provide you habitable furnishings. And perhaps your locality has some regulation regarding that. Otherwise, it may be vague. Common sense may come into play here. A week would be too long. Six hours might be unreasonably short. A day or two is perhaps in the realm of reason.
What can you do if it's not done? That depends on your lease. If the lease says it's the responsibility of the landlord and the landlord refuses to fix it, then you could argue that your home is no longer habitable and that the landlord has broken the lease. However, have a lawyer advise you on this, and what steps to take.
Realistically: Notify the landlord as soon as possible. And it's possible that your renter's insurance might cover the loss of the contents of the refrigerator.
Hope that helps.
There should be a section in your lease that discusses maintenance/ repairs and you need to read that.
If there is a property management company that you pay your rent to .. then they are responsible for those repairs.
If you pay the landlord directly for the rent and you cannot find any repair info on your lease, then most likely the landlord has the duty to handle your repairs.