$HoHoH->{"alice"}{"aaaa"} = "20:00:00";
# instead of
# $HoHoH->{"alice"}{"aaaa"}{"time"} = "20:00:00";

##</code><code>##

foreach my $user ( sort keys %$HoHoH ) {
  print "$user\n";
  for my $text
      ( sort {$HoHoH->{$user}->{$a}
              cmp
              $HoHoH->{$user}->{$b}
             }
       keys %{$HoHoH->{ $user }} ) {
...