Here is a proof following <PERSON>, "Lorentz Invariance vs. Temperature in QFT", Letters in Mathematical Physics (1986) Vol. 11, Issue 1 (1986) 73-80. The first two pages of the paper are available for free here, but the website wants money for more of the paper. (Click the orange "Look Inside" button if the paper doesn't open automatically.) Fortunately, the proof is on the second page.
Define $$w(A) = tr\bigl(e^{-\beta H} A\bigr)/tr\bigl(e^{-\beta H}\bigr).$$ The KMS condition can be written $$w(\phi(x)\phi(y)) = w(\phi(y)\phi({\tilde x}))$$ where $\tilde x$ is $x$ with the time component shifted by $i\beta$.
Now consider the Fourier transform $$\langle\phi_k\phi_{-k}\rangle = \int d^4x d^4y\ e^{i k \cdot (x-y)} w(\phi(x)\phi(y)).$$ By the KMS condition this is $$\langle\phi_k\phi_{-k}\rangle = \int d^4x d^4y\ e^{i k \cdot (x-y)} w(\phi( y)\phi({\tilde x})).$$ Shifting the time in the $x$ integral, this becomes $$e^{\beta k_0}\int d^4{\tilde x} d^4 y\ e^{i k \cdot ({\tilde x}- y)} w(\phi(y)\phi({\tilde x})) = e^{\beta k_0}\langle\phi_{-k}\phi_{k}\rangle,$$ so we have $$\langle\phi_k\phi_{-k}\rangle = e^{\beta k_0}\langle\phi_{-k}\phi_{k}\rangle.$$ Starting with the right hand side of this equation, if Lorentz invariance holds, $\langle\phi_{-k}\phi_{k}\rangle$ is a scalar under <PERSON> transformations for a scalar field $\phi$. 592
However, $e^{\beta k_0}$ is manifestly not a <PERSON> scalar since it depends non-covariantly on $k_0$. This implies that the left hand side of the above equation is not a Lorentz scalar, contradicting the Lorentz invariance of $\langle\phi_k\phi_{-k}\rangle$.
This proves that Lorentz covariance cannot hold for finite $\beta$.
Another way to see that Lorentz covariance is broken at finite temperature is to Wick rotate to Euclidean spacetime. The KMS condition then implies periodicity of the Green's function in the time direction. Lorentz transformations in real spacetime are mapped to rotations in Euclidean spacetime by the Wick rotation. Since periodic boundary conditions are imposed in one direction and not the other three directions, rotation symmetry is broken in Euclidean spacetime and therefore Lorentz symmetry is broken in real spacetime. 469