What causes hyundai Elantra 2002 tempature gauge fluctuates between normal and hot

The thermostat "seals" the cool coolant giving up (rejecting) heat in the radiator & heater core from the hot coolant absorbing heat in the running engine "block".

When the coolant in the engine reaches a specific temperature (180 ~ 195 degrees) as determined by the value selected by manufacturer or when replaced, it opens and allows the hot coolant in the engine block to be pumped out to the radiatot & heater core for cooling and the cool coolant in the radiator / heater core is pumped into the engine block - replacing the hot coolant. The thermostat senses cool coolant in the engine block and seals shut until the temperature of the coolant in the engine block rises to 180 ~ 195 degrees again. This process continues over and over for as long as needed.

You should check to make sure that the coolant level in the reservoir / radiator is sufficient by checking when cold. Do NOT remove a radiator cap when it is hot! With the cap off, start the engine and add additional coolant as the engine warms up to normal operating temperature. If coolant levels are ok, the trouble could be related to the thermostat or possibly a faulty temperature sending switch. This means that the temperature of the coolant is fine - but the switch or sensor that converts the temperature to a voltage for the gauge on yur dashboard is the problem.

If the temperature is fluctuating up and down, you should have it fixed. Good luck!