  We report a Giant Metrewave Radio Telescope (GMRT) survey for associated HI
21-cm absorption from 50 active galactic nuclei (AGNs), at $z \approx 0.04 -
3.01$, selected from the Caltech-Jodrell Bank Flat-spectrum (CJF) sample. Clean
spectra were obtained towards 40 sources, yielding two new absorption
detections, at $z = 0.229$ towards TXS 0003+380 and $z = 0.333$ towards TXS
1456+375, besides confirming an earlier detection, at $z = 1.277$ towards TXS
1543+480. There are 92 CJF sources, at $0.01 \lesssim z \lesssim 3.6$, with
searches for associated HI 21-cm absorption, by far the largest
uniformly-selected AGN sample with searches for such absorption. We find weak
($\approx 2\sigma$) evidence for a lower detection rate of HI 21-cm absorption
at high redshifts, with detection rates of $28^{+10}_{-8}$\% and
$7^{+6}_{-4}$\% in the low-$z$ ($z < z_{\rm med}$) and high-$z$ ($z > z_{\rm
med}$) sub-samples, respectively. We use two-sample tests to find that the
strength of the HI 21-cm absorption in the AGNs of our sample depends on both
redshift and AGN luminosity, with a lower detection rate and weaker absorption
at high redshifts and high ultraviolet/radio AGN luminosities. Unfortunately,
the luminosity bias in our sample, with high-luminosity AGNs arising at high
redshifts, implies that it is not currently possible to identify whether
redshift evolution or AGN luminosity is the primary cause of the weaker
absorption in high-$z$, high-luminosity AGNs. We find that the strength of HI
21-cm absorption does not depend on AGN colour, suggesting that dust extinction
is not the main cause of reddening in the CJF sample.
