MWD (Measurement While Drilling) or LWD (Logging While Drilling) will give you a continuous borehole temperature profile in real-time among many other measurements.
I think that the question needs qualification. The bit is composed of a number of elements, and obviously these will all have quite different temperature exposures. You can imagine quite easily that the cutting edges of the PDC elements will be different to the pin. And a roller bit with long teeth will be different to a .... etc. So: try to tell us what exactly you're looking for ... and why.