I can offer some general information.
The rights and obligations of both landlords and tenants are typically governed by local tenancy laws and the terms outlined in the lease agreement or rental contract. These laws can vary significantly depending on your country, state, or jurisdiction.
In most cases, a landlord cannot dictate how you use the utilities within your rental unit as long as you are not violating any local laws or causing damage to the property. If there is no specific clause in your lease agreement that states you must keep your balcony door shut while running the AC, then your landlord may not have the authority to demand such a requirement.
To get a definitive answer on your specific situation, you should review your lease agreement thoroughly, including any additional addendums or rules provided by the landlord. If you are still uncertain or need clarification, it's best to consult with a local legal professional who specializes in tenancy or landlord-tenant law. They can provide you with accurate advice based on the laws in your area and the specific details of your lease agreement.