# you initialise/calculate $url here

if ($ENV{'HTTP_USER_AGENT'} =~ /facebookexternalhit/) 
{
   # you only need $url here
   print "Content-Type: text/html\n\n";
print <<EOM;
$url
EOM
}

else {
  # you don't need $url here
print "Location: http://www.youtube.com/watch?v=$video\n\n";
EOM
}
[download]

You are effectively doing a HTTP GET for every request you get, facebook bot or not.