We report results from a programme aimed at investigating the temperature of neutral gas in high-redshift damped Lyman α absorbers (DLAs). This involved (1) H i 21 cm absorption studies of a large sample of DLAs towards radio-loud quasars, (2) very long baseline interferometric studies to measure the low-frequency quasar core fractions, and (3) optical/ultraviolet spectroscopy to determine DLA metallicities and the velocity widths of low-ionization metal lines. Including literature data, our sample consists of 37 DLAs with estimates of the harmonic mean spin temperature T s . We find a statistically significant (4) difference between the T s distributions in the high- z ( z 〉 2.4) and low- z ( z 〈 2.4) DLA samples. The high- z sample contains more systems with high spin temperature, T s 1000 K. The T s distributions in DLAs and the Galaxy are also significantly (6) different, with more high- T s sightlines in DLAs than in the Milky Way. The high T s values in the high- z DLAs of our sample arise due to low fractions of the cold neutral medium (CNM). Only 2 of 23 DLAs at z 〉 1.7 have T s values indicating CNM fractions 〉20 per cent, comparable to the median value (27 per cent) in the Galaxy. We tested whether the H i column density measured towards the optical quasar might be systematically different from that towards the radio core by comparing the H i column densities inferred from H i 21 cm emission studies at different spatial resolutions (15 pc-1 kpc) in the Large Magellanic Cloud. The high-resolution N H i values are, on average, larger than the smoothed ones for N H i 〉 10 21 cm –2 , but lower than the smoothed N H i estimates for N H i 〈 10 21 cm –2 . Since there are far more DLAs with low N H i values than high ones, the use of the optical N H i value for the radio sightline results in a statistical tendency to underestimate DLA spin temperatures. For 29 DLAs with metallicity estimates, we confirm the presence of an anticorrelation between T s and metallicity [ Z /H], at 3.5 significance via a non-parametric Kendall-tau test. This result was obtained with the assumption that the DLA covering factor is equal to the core fraction. However, Monte Carlo simulations show that the significance of the result is only marginally decreased if the covering factor and the core fraction are uncorrelated, or if there is a random error in the inferred covering factor. We also find statistically significant evidence for redshift evolution in DLA spin temperatures even for the DLA sub-sample at z 〉 1. Since all DLAs at z 〉 1 have angular diameter distances comparable to or larger than those of their background quasars, they have similar efficiency in covering the quasars. We conclude that low covering factors in high- z DLAs cannot account for the observed redshift evolution in spin temperatures.