Is it permissible for the landlord to take rent for the remaining period of the contract after the apartment has been rented to someone else, and what is the ruling on lying to them to get out of the apartment?

The question addresses three issues: the ruling on lying, the cancellation of a lease, and the right of the apartment owners to the rent after the tenant vacates. Lying is a sin that a believer does not engage in except in specific cases, such as reconciling people, during war, or between spouses. What you did does not fall into these categories, so you must repent from it. Fulfilling contracts is obligatory, and a lease is a binding contract that cannot be canceled except by mutual consent. Since you canceled the contract through deception, the benefit of the apartment reverts to its owners. Part of your complete repentance is to pay the apartment owners the difference between what you were supposed to pay and what they received from the new tenant, or to seek their forgiveness.