We charge a fixed shipping rate for subtotal under a certain amount, and offer free shipping to orders over this threshold.
To achieve that, initially I created a shopping cart price rule in Magento promotions, and specified free shipping conditions and actions. Then I enabled Flatrate as a shipping method. This shipping method working in conjunction with the promotion rule served me well until I find it does not work for Google Checkout as a payment method.
Magento Googlecheckout always charges that fixed shipping rate regardless of the promotion rules. I assume it is a bug. To avoid the bug, I delete the free shipping promotion rule and enable Tablerates to replace Flatrate. I can not fix this bug. The bug exists in Magento 18.104.22.168. Giving time it might be fixed by Magento core team.