What is the Unruh effect for an accelerated observer in heat bath? The Unruh effect states an accelerated observer in flat Minkowski spacetime sees excited states in a heat bath with the Unruh temperature.
Then, when the initial rest observer is in a heat bath with the uniform temperature $T$ in the Minkowski spacetime, what is the temperature the observer will see after some acceleration? Is it $T+T_{\mathrm{Unruh}}$?
 A: Because the velocity of the observer has no effect on the temperature of the heat bath (reservoir) or, the other way round the temperature of a get bath is not dependent on the velocity of the bath, the answer is yes.
Of course at relativistic speeds, the mass of the bath will increase. But one can always find a reference frame in which it's velocity is zero. If the frame is accelerating together with the observer, the bath will appear to be in a gravitational field. This too will have no influence on the temperature.
Note that The Minkowskski spacetime is used mostly in SR since it has no intrinsic curvature. Puting different M-spaces together (for a mass with a continuously varying velocity) creates a curved spacetime.
This is essentially the same as asking if the temperature of a box filled a gas changes when it moves. The answer to this question is no.
Only for enormous accelerations, the Unruh effect will cause a change in temperature for the observer measuring the temperature.

an accelerating thermometer (like one being waved around) in empty space, subtracting any other contribution to its temperature, will record a non-zero temperature, just from its acceleration

Taken from this Wikipedia article. In this article, it is also stated:

It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are disputed

