Lighttpd

lighttpd/1.4.69 (ssl) - a light and fast webserver

You activate dir-listing only for url-paths matching ^/t/

$HTTP["url"] =~ "^/t/" {
     dir-listing.activate = "enable" 
     dir-listing.show-size = "enable" 
}

Make sure you clear your browser cache. Try a different browser, or try on a different machine.

I recommend testing with a command line tool such as curl and looking at the HTML to verify that the HTML contains the reference to the external CSS file.

Thanks for the reply.
The problem was <link> tag's path in 'href' attribue it must be relative to the location of the page. Now it is as follows and everything works ok.

<link rel="stylesheet" type="text/css" href="../css/custom-dir.css">

Yes. dir-listing.external-css is a url-path, and can be root-relative if the .css is on the same site, or can be a different URL. As documented on mod_dirlisting

dir-listing.external-css

URL to an external css stylesheet for the directory listing.

Wrong: dir-listing.external-css = "/var/www/html/css/custom-dirlisting.css"
Correct: dir-listing.external-css = "/css/custom-dirlisting.css"
Correct: dir-listing.external-css = "https://example.com/css/custom-dirlisting.css"

Tested. This is correct.
Thank you.