Update notifier function for a custom WordPress Theme

update.json

{
    "new_version":"1.0.0",
    "url":"http://www.domain.com/theme-name/changelog",
    "package":"http://www.domain.com/theme-name/themes.zip"
}

wp-theme-update-notifier.php

<?php
/**
 * Theme update notifier
 *
 * @author Luc Poupard
 *
 * Based on Xpark Media code :
 * @link https://xparkmedia.com/blog/add-update-notification-selfhosted-premium-themes-plugins/
 * But using wp_remote_get() instead of curl_init() :
 * @link https://pippinsplugins.com/using-wp_remote_get-to-parse-json-from-remote-apis/
 */

function theme_notifier ( $transient ) {
    if( empty( $transient->checked['theme-name'] ) )
    return $transient;
    
    $response = wp_remote_get( 'http://www.domain.com/theme-name/update.json' );
    if( is_wp_error( $response ) ) {
        return false;
    }
    
    $result = wp_remote_retrieve_body( $response );
    $data = json_decode( $result );
    
    if( ! empty( $data ) ) {
        if( version_compare( $transient->checked['theme-name'], $data->new_version, '<' ) )
        $transient->response['theme-name'] = (array) $data;
    }
    
    return $transient;
}
add_filter ( 'pre_set_site_transient_update_themes', 'theme_update_notifier' );