The EVSE (Electric Vehicle Supply Equipment) doesn't "charge" the battery. Each EV has an onboard AC to DC converter and battery charger that monitors the pack voltage and temperature with built-in sensors until its limits have been reached. Then the charger tells the EVSE to stop the AC supply. In few words, the EVSE (public or home installed) is an intelligent "extensionn" that communicates with the EV onboard charger using the SAE J1772 standard to prevent harm to the vehicle or the owner.
The onboard charger is what determines if the pack need a "trickle charge", and may cycle the EVSE on and off to supply AC power as needed. The pack temperature versus ambient temperature management is one arrangement where the EVSE may indeed cycle on and off to maintain an established battery temperature using the EV's built-in heating or cooling systems, and you willl get charged for its supply while connected to the EVSE.
In the future, when we have larger packs, probably over 40 kWh, we will need fast DC chargers and those will have their intelligence offboard, not in the EV, but connected to the same sensors inside the EV pack, because that external charger bypasses the smaller onboard charger.