What is fetching your $robots_txt file?  As far as I understand, WWW::RobotRules does not fetch files itself, but rather expects you to use some user agent like LWP::Simple or LWP::UseAgent for that...  And with the latter, you could then just call the timeout method on the user agent object, e.g.

use LWP::UserAgent;
my $ua = LWP::UserAgent->new;
$ua->timeout(2);
...
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or, when you're using LWP::Simple, you have to import $ua to get access to its internal LWP::UserAgent object:

use LWP::Simple qw(get $ua);
$ua->timeout(2);
...
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Or better yet, simply use LWP::RobotUA in the first place, which should provide the same timeout method, as it's a sub-class of LWP::UserAgent.  Something like this (if I'm reading the docs correctly — i.e. untested):

use LWP::RobotUA;
use WWW::RobotRules;

my $ua = LWP::RobotUA->new(...);
my $rules = WWW::RobotRules->new(...);
$ua->rules($rules);  # optional - defaults used otherwise
$ua->timeout(2);

my $response = $ua->get(...);
...
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