Bigian's answer is probably the best for this, but another way to speed things up is to use LWP :: ConnCache to allow LWP to reuse existing connections, rather than creating a new connection for each request.
Enabling this is simple if you only knock on one site -
my $conn_cache = LWP::ConnCache->new;
$conn_cache->total_capacity([1]) ;
$ua->conn_cache($conn_cache) ;
, http- https.