When someone gives notice to break a lease and the landlord rents the apartment immediately, can the landlord charge rent to both parties?

I don't have time today to look up FL tenant law. Most states allow charges for money and time spent on finding a new tenant, but do not allow to charge for the rest of the rental term, once a new tenant is found.

Keep in mind, however, that no state allows for a tenant to 'give notice' to break a lease, unless it's a month-to-month lease, i.e., a tenant at will, in which either party can give a month's notice.

Agreed, you can not 'give notice" to break a lease, you may tell them and you may break it.
They may recoup all costs of cleaning, advertising, and costs. They can not collect rent for the same apartment twice. After they are satisfied of their costs.

Read your lease agreement. If you were breaking a lease, just moving out without proper notice, unless there was dire need, I would keep your last months rent. They can always have need for fixup and repair. Security deposit?

A lot of places REQUIRE a repaint before new tenants move in... for people that honor their lease its part of doing business... when people break their lease that imposes an extra unexpected expense for the landlord.. which along with damages... they CAN collect for.