@Sven ,
Thank you for reaching out to the Microsoft Q&A platform. Happy to answer your question.
Changing the default access tier setting for a storage account applies to all blobs in the account for which an access tier has not been explicitly set. If you toggle the default access tier setting from Hot to Cool in a general-purpose v2 account, then you are charged for write operations (per 10,000) for all blobs for which the access tier is inferred. The destination tier in your case was cool tier and hence you were charged with Cool Write Operations for hot->cool configuration change. For more information, please read the article.
https://learn.microsoft.com/en-us/azure/storage/blobs/storage-blob-storage-tiers#blob-level-tiering-billing
Thanks
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