We present deep GMRT 21cm absorption spectra of 10 damped Lyman- \alpha systems ( DLAs ) , of which 8 are at redshifts z \ga 1.3 . HI absorption was detected in only one DLA , the z = 0.5318 absorber toward PKS 1629+12 . This absorber has been identified with a luminous spiral galaxy ; the spin temperature limit ( T _ { s } \leq 310 K ) derived from our observations continues the trend of DLAs associated with bright spirals having low spin temperatures . In seven of the remaining 9 systems , the observations place strong lower limits on the spin temperature of the HI gas . We combine this sample with data taken from the literature to study the properties of all known DLAs with 21cm absorption studies . The sample of DLAs which have been searched for 21cm absorption now consists of 31 systems , with T _ { s } estimates available in 24 cases ; of these , 16 are at z < 2 and 8 at z > 2 , with 11 ( all at z < 1 ) having optical identifications . For the latter 11 DLAs , we find that all of the low T _ { s } DLAs have been identified with large , luminous galaxies , while all the DLAs with high spin temperature ( T _ { s } \ga 1000 K ) have been identified either with LSBs or dwarfs . Further , we find no correlation between impact parameter and spin temperature ; it is thus unlikely that the high measured T _ { s } values for DLAs arise from lines of sight passing through the outskirts of large disk galaxies . Instead , the spin temperature of DLAs appears to correlate with the host galaxy type . The trend ( noted earlier by Chengalur & Kanekar 2000 ) that low z DLAs exhibit both high and low T _ { s } values while high redshift ( z \ga 3 ) DLAs only show high spin temperatures is present in this expanded data set . Based on this difference in spin temperatures , the Gehan test rules out the hypothesis that DLAs at z > 2 and DLAs at z < 2 are drawn from the same parent population at \sim 99 % confidence level . Finally , we use the new GMRT spectra along with 2 spectra from the literature to estimate upper limits on the fraction of cold HI , f _ { CNM } , in DLAs at z \ga 3 . For local spirals , f _ { CNM } \sim 0.5 ; in contrast , we find that f _ { CNM } < 0.3 in all 7 high z absorbers , and f _ { CNM } < 0.1 in 5 of the 7 cases .