Is temperature a Lorentz invariant in relativity?

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.

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.

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.