If an observer starts moving at relativistic speeds will he observe the temperature of objects to change as compared to their rest temperatures?
Suppose the rest temperature measured is $T$ and the observer starts moving with speed $v$. What will be the new temperature observed by him?
This is a very good question. Einstein himself, in a 1907 review (available in translation as Am. J. Phys. 45, 512 (1977), e.g. here), and Planck, one year later, assumed the first and second law of thermodynamics to be covariant, and derived from that the following transformation rule for the temperature:
$$
T' = T/\gamma, \quad \gamma = \sqrt{1/(1-v^2/c^2)}.
$$
So, an observer would see a system in relativistic motion "cooler" than if he were in its rest frame.

However, in 1963 Ott (Z. Phys. 175 no. 1 (1963) 70) proposed as the appropriate transformation
$$
T' = \gamma T
$$ 
suggesting that a moving body appears "relatively" warmer.

Later on Landsberg (Nature 213 (1966) 571 and 214 (1967) 903) argued that the thermodynamic quantities that are statistical in nature, such as temperature, entropy and internal energy, should not be expected to change for an observer who sees the center of mass of the system moving uniformly.
This approach, leads to the conclusion that some thermodynamic relationships such as the second law are not covariant and results in the transformation rule:
$$
T' = T
$$

So far it seems there isn't a general consensus on which is the appropriate transformation, but I may be not aware of some "breakthrough" experiment on the topic.

Main reference: 


  M.Khaleghy, F.Qassemi. Relativistic Temperature Transformation Revisited, One hundred years after Relativity Theory (2005). arXiv:physics/0506214.

