The hotplate light on my stove
That is probably normal (with a small possibility of being a defect).  The hot indicator is precisely that, and is designed to turn off only after all spots have definitely cooled.  

And it may not know when that is, because it may be a simple timer, rather than a temperature sensor.  And it may be designed for worst case, i.e. A pot left on the burner for quite some time after it's shut off, and thus retaining a lot of heat. 

You might think the light would only come on when power is switched to the burner.  But the above feature is cheap to add, and provides a big safety improvement, so it wouldn't surprise me if a standards body like NFPA or UL made it a requirement.  Of course that feature is also more complex, so more likely to malfunction.