Yes, on some vehicles it can.
What is a thermostat light?
The side of the thermostat that has a spring showing faces into the engine.
Not sure if by your question you mean the thermostat opens slow and the engine overheats or if you mean the engine takes a long time to warm up. If the engine is overheating, more than likely the thermostat is stuck in the closed position and needs to be replaced before engine damage is caused because of overheating. If the engine is taking a long time to warm up, the thermostat is probably stuck in the open position and has to be replaced so the engine can reach its normal operating temperature in a shorter time.
Front right of the engine.
With the spring part inside the engine
The spring goes down into the engine.
Probably at the engine end of the upper radiator hose
On my 95 pickup with 4 cylinder engine, the thermostat is located where the lower radiator hose enters the engine block. Hope this helps.
Need to know engine size and year.
It is on the left or right side of the truck by the fender well on the engine.
Under the housing where the upper radiator hose hooks to the engine.
There is something not right with the engine.