Good idea, some quick googling found this post by Graham Knop from which I adapted the code:

use warnings;
use strict;
use LWP::UserAgent;

my $ua = LWP::UserAgent->new(
    ssl_opts => {
        SSL_verify_callback => sub {
            my ($ok, $ctx_store) = @_;
            my $cert = Net::SSLeay::X509_STORE_CTX_get_current_cert($c
+tx_store);
            print Net::SSLeay::P_ASN1_TIME_get_isotime(
                Net::SSLeay::X509_get_notAfter($cert) ), "\n";
            return $ok;
        },
    },
);
$ua->get('https://www.perlmonks.org/');

__END__

2038-01-18T23:59:59Z
2029-08-21T23:59:59Z
2022-09-02T23:59:59Z
[download]

Note the documentation also says "The callback will be called for each element in the certificate chain."

Another post in the above thread points to Net::SSL::ExpireDate.