Doubling CO2 will cause approximately 2-4°C increase in annual mean surface air temperature. That means that Svensmark's claim that the increase would be less than this is false.
The last two assessment reports from the Intergovernmental Panel on Climate Change (IPCC) agree that the temperature increase of doubling CO2 is something around the alleged. Model runs from their fourth assessment report (AR4) estimates it at 2.1-4.4°C while the fifth assessment report (AR5) estimates it at 2.1-4.7°C. The ensemble mean is 3.2°C in both cases. (AR5, Section 9.7.1)
The overall assessment of the Equilibrium Climate Sensitivity (ECS) according to (AR5, Section 10.8.2) is:
...there is high confidence that ECS is extremely unlikely less than 1°C and medium confidence that the ECS is likely between 1.5°C and 4.5°C and very unlikely greater than 6°C.
However this value is not for each doubling of CO2, but for the first doubling from the pre-industrial levels. For the next doubling the value will probably be different.
The AR5 says specifically about the proposed link (AR5, Section 7.4.6):
Correlations between cosmic ray flux and observed aerosol or cloud properties are weak and local at best, and do not prove to be robust on the regional or global scale. Although there is some evidence that ionization from cosmic rays may enhance aerosol nucleation in the free troposphere, there is medium evidence and high agreement that the cosmic ray-ionization mechanism is too weak to influence global concentrations of CCN or droplets or their change over the last century or during a solar cycle in any climatically significant way.