We present a comprehensive analysis of the spin temperature/covering factor degeneracy, T spin /f , in damped Lyman-α absorption systems. By normalising the upper limits and including these via a survival analysis, there is, as previously claimed, an apparent increase in T spin /f with redshift at z abs > ∼ 1. However, when we account for the geometry effects of an expanding Universe, neglected by the previous studies, this increase in T spin at z abs > ∼ 1 is preceded by a decrease at z abs < ∼ 1. Using high resolution radio images of the background continuum sources, we can transform the T spin /f degeneracy to T spin /d 2 abs , where d abs is the projected linear size of the absorber. Again, there is no overall increase with redshift, although a dip at z abs ≈ 2 persists. Furthermore, we find d 2 abs /T spin to follow a similar variation with redshift as the star formation rate, ψ * . This suggests that, although the total hydrogen column density, N HI , shows little relation to ψ * , the fraction of the cold neutral medium, τ obs dv/N HI , may. Therefore, further efforts to link the neutral gas with the star formation history should also consider the cool component of the gas.