Is temperature a Lorentz invariant in relativity? 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?
 A: This article from 2017 gives a nice overview of the topic at a level intended to be accessible to undergraduate physics majors.
What is the temperature of a moving body?
by Cristian Farías, Victor A. Pinto & Pablo S. Moya

The construction of a relativistic thermodynamics theory is still
controversial after more than 110 years. To the date there is no
agreement on which set of relativistic transformations of
thermodynamic quantities is the correct one, or if the problem even
has a solution. Starting from Planck and Einstein, several authors
have proposed their own reasoning, concluding that a moving body could
appear cooler, hotter or at the same temperature as measured by a
local observer. In this article we present a review of the main
theories of relativistic thermodynamics, with an special emphasis on
the physical assumptions adopted by each one.

A: One thing to note is observing something's temperature and thermodynamic notions of temperature aren't exactly the same thing. This is in line with @Mattia 's answer. If a star is receding form you then it will appear cooler because its radiation has been red-shifted. Does this mean that there can be a net flow of heat from us to the star (provided it's moving fast enough)? In the rest frame of the star, our radiation is red-shifted, so this would lead to a paradox.
On the other hand, for accelerating observers there is what's known as Unruh radiation, very much analogous to Hawking radiation. An accelerated observer appears to be radiating energy as though it has been heated, and in its own frame, observes the vacuum to have a thermal spectrum. Since there is acceleration, there is no requirement of thermal equilibrium.
A: Loooking at a mole of an ideal gas you can deduce that if there ARE any consistent transformations of the thermodynamic state variables the transformation of the product $k·T$ is given by  $k'·T' = k·T/\gamma$.
Planck (and others) opted for $k' = k$ (but his proof for this 'begs the question' !).
There are very good arguments for  $T' = T$ and hence  $k' = k/\gamma$ . The main theorems of thermodynamics are form invariant.
$R = k·N_{A} = P_{0}·V_{0}/T_{0}$ can only be invariant if temperatures transform in the same way as volumes do, that is by multiplying by the root.
All the details and references are to be found in
http://www.physastromath.ch/uploads/myPdfs/Relativ/T_SRT_en.pdf
A: Cubero et al. 2007: Thermal equilibrium and statistical thermometers in special relativity (http://arxiv.org/abs/0705.3328) came to the conclusion 

that 'temperature' can be statistically defined and measured in an
  observer frame independent way.

With fully relativistic 1D molecular dynamics simulations they verified that the temperature definition given by Landsberg Nature 214 (1967) 903) defines a Lorentz invariant gas thermometer on a purely microscopic basis.
A: The answer to this long standing question has been given by Landsberg. But it seems this answers was overlooked by many (including myself, see my wrong answer here).
There is no universal relativistic temperature transformation of the form $T' = T(v)$ . 


*

*Landsberg (1996): Laying the ghost of the relativistic temperature
transformation 

*Landsberg (2004): The impossibility of a
universal relativistic temperature transformation
Why? Let's look at the example of a moving black body. The black body spectrum of a moving black body shows a frequency shift due to the relativistic doppler effect. The doppler effect however depends on the angle $\alpha$ between observer and the source. This leads effectively to an angle dependent temperature for a moving black body:
$$
T'(\alpha, v) = \frac{T \sqrt{1-\frac{v^2}{c^2}}}{1 - \frac{v}{c} \cos \alpha }
$$
(see e.g. https://en.wikipedia.org/wiki/Black-body_radiation#Doppler_effect_for_a_moving_black_body)
So an observer
moving in a heat reservoir cannot detect an isotropic blackbody spectrum and hence cannot find a parameter which can be identified as temperature.
This is an important effect in astronomy. For example the cosmic microwave background shows a temperature anisotropy due to the movement of the earth relative to the background, a fact which has been explicitly calculated in the 60s, e.g.


*

*Heer (1968): Theory for the Measurement of the
Earth’s Velocity through the 3°K Cosmic Radiation

*Henry (1968): Distribution of Blackbody Cavity Radiation in a Moving Frame of Reference
But as Landsberg also notes they basically just rediscovered what Pauli had already published in his famous article/book about the black body radiation in a moving frame of reference:


*

*Pauli, Die Relativitätstheorie, 1921, Encycl. d. Math. Wiss, Vol. 5 (Leipzig: Teubner) p. 698,
translated as Theory of Relativity, London, Pergamon, 1958.
A: 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.

A: Suppose a mercury thermometer is prepared such that its bulb is in contact with a heat source at temperature T. The length of the responding mercury column is L. Now, imagine that the bulb defines the origin of coordinates of a lab frame such that the thermometer lies on its x-axis with +L as the coordinate of the column end. A relativistic observer moving along the x-axis measures the length of the column  Evidently, that observer would measure the Lorentz contracted length L/gamma, and thus, relative to an identical thermometer set-up in his frame, would infer a temperature Tob = T/gamma. 
However, from a purely thermodynamic point of view, the temperature of one body cannot be registered by another (say a thermometer) unless those bodies are in a thermal contact that allows a small amount of heat to be absorbed by the thermometer. Moreover, starting from its first contact with the thermometer, the reading cannot occur until thermal equilibrium is established.
It thus seems that the thought experiment above is the wrong set-up because the bulb of the observer thermometer must be dipped into the lab frame heat bath as it goes by. Assuming a big lab system so that enough time has passed for the two systems to come into thermal equilibrium, they would be at the same temperature. 
Apparently the temperature is a quantity that evolves into a Lorentz scalar through the establishment of thermal equilibrium.
