Yes, in the United States, it is legal to rent an apartment without air conditioning in most cases. There are no federal laws that require rental properties to have air conditioning. Air conditioning is considered an amenity or feature that some apartments may offer, but it is not a mandatory requirement for landlords.
However, certain states, cities, or local jurisdictions might have specific housing codes or regulations that mandate certain living conditions or amenities for rental properties. For instance, in some areas with extreme heat, there might be local regulations that require landlords to provide air conditioning or proper ventilation in rental units. These regulations vary by location, so it's essential to check the specific laws and regulations in the area where you are looking to rent.
If you are concerned about the availability of air conditioning in a rental property, it's a good idea to ask the landlord or property management company about the amenities included in the apartment before signing the lease. If air conditioning is crucial to you, you may want to focus your search on apartments that explicitly advertise air conditioning as one of their features or amenities.