that happens because the relay is cut off relay - meaning when the relay works the fan switches off, so when the ignition is turned on the relay switches on and the relay coil remains energized. It only gets cut off when the thermoswitch or A/C high pressure trips and causes the fan(s) to run.
Dont worry its normal operation, the relay can take it. however later toyotas came equipped with German made Bosch "cut off relay" its far more robust than the POS denso round relay.
You might wonder why the fans dont run when the key is turned off because the relay is ON at that point. Your toyota also has a main relay that grounds all hot points when you switch off the ignition key. It creates massive headaches to someone who does not know of this function.
People might think its stupid to design it like this but consider this if the wire to the sensor breaks or the wiring gets damaged somewhere within the wiring loom the fan will run constantly, unlike if a regular relay setup that would not switch on the fan when the wiring gets damaged and overheat the engine to death.