  We constrain deviations of the form $T\propto (1+z)^{1+\epsilon}$ from the
standard redshift-temperature relation, corresponding to modifying distance
duality as $D_L=(1+z)^{2(1+\epsilon)} D_A$. We consider a consistent model, in
which both the background and perturbation equations are changed. For this
purpose, we introduce a species of dark radiation particles to which photon
energy density is transferred, and assume $\epsilon\ge0$. The Planck 2015
release high multipole temperature plus low multipole data give the limit
$\epsilon<4.5\times 10^{-3}$ at 95% C.L. The main obstacle to improving this
CMB-only result is strong degeneracy between $\epsilon$ and the physical matter
densities $\omega_{\rm b}$ and $\omega_{\rm c}$. A constraint on deuterium
abundance improves the limit to $\epsilon<1.8\times 10^{-3}$. Adding the Planck
high-multipole CMB polarisation and BAO data leads to a small improvement; with
this maximal dataset we obtain $\epsilon<1.3\times 10^{-3}$. This dataset
constrains the present dark radiation energy density to at most 12% of the
total photon plus dark radiation density. Finally, we discuss the degeneracy
between dark radiation and the effective number of relativistic species $N_{\rm
eff}$, and consider the impact of dark radiation perturbations and allowing
$\epsilon<0$ on the results.
