Thanks to both of you.
I installed and activated wp-cron.php and wp-cron-future-pings.php but the ping still comes when I publish even though I scheduled the post to tomorrow morning.
I also installed and activated wp-cron-reminder and got 8 60 minute reminders in less than 5 minutes.
I know virtually nothing about php and related stuff. But I did look through wp-cron.php and saw that on row 24 when you get the hour you use ‘h’. On my blog I use ‘H’ to get the 24-hour format instead of 12-hour format since we use 24h here in Sweden. May that be the problem?