  (Context) We calculate constraints from current and future cosmic microwave
background (CMB) measurements on annihilating dark matter (DM) with masses
below the electroweak scale: m_{DM}=5-100 GeV. In particular, we focus our
attention on the lower end of this mass range, as DM particles with masses
m_{DM} ~ 10 GeV have been recently claimed to be consistent with the CoGeNT and
DAMA/LIBRA results, while also providing viable DM candidates to explain the
measurements of Fermi and WMAP haze. (Aims) We study the model (in)dependence
of CMB spectrum on particle physics DM models, large scale structure formation
and cosmological uncertainties. We attempt to find a simple and practical
recipe for estimating current and future CMB bounds on a broad class of DM
annihilation models. (Results) We show that in the studied DM mass range the
CMB signal of DM annihilations is independent of the details of large scale
structure formation, distribution and profile of DM halos and other
cosmological uncertainties. All particle physics models of DM annihilation can
be described with only one parameter, the fraction of energy carried away by
neutrinos in DM annihilation. As the main result we provide a simple and rather
generic fitting formula for calculating CMB constraints on the annihilation
cross section of light WIMPs. We show that thermal relic DM in the CoGeNT, DAMA
favoured mass range is in a serious conflict with present CMB data for the
annihilation channels with few neutrinos, and will definitely be tested by the
Planck mission for all possible DM annihilation channels. Also, our findings
strongly disfavor the claim that thermal relic DM annihilations with m_{DM} ~
10 GeV and $<sigma_av> ~ 9x10^{-25} cm^{3}s^{-1} could be a cause for Fermi and
WMAP haze.
