In the paper Quantum field aspect of Unruh problem (and others with similar approaches) Buchholz and Verch show that applying the rigorous algebraic approach to QFT, the derivation of the Unruh effect usually done by almost everyone is incorrect, and that the Unruh effect doesn't exist.
More than that, as far as I know, the Unruh effect hasn't been observed. So in the end of the day, the Unruh effect has no mathematicaly rigorous theoretical derivation, nor any kind of experimental observation that would point towards its correctness.
Either way, lots of physicists insist the effect exists and claim that the "mathematicaly rigorous approach thing" is nonsense.
Well, if the prediction was actually observed (as many things in QFT which are not mathematicaly rigorous), I could agree that probably the predicition is correct and certainly one day a rigorous mathematical approach will be found. The issue here is that the prediction simply isn't observed.
Why so many physicists believe the Unruh efffect exist then, when it has been established the traditional derivation is simply plain wrong and there is no experimental evidence of the effect?
Edit: Regarding more material on the subject, there are some more recent papers:
The first one is from 2015 and it seems it has been published. The second article is from this year, but I think it hasn't been published yet (although as far as I know it will be). From a mathematical point of view I've found the second paper quite convincing. The might be more that I don't know about yet.