Beck234 Greetings & Welcome to Microsoft Q&A forum!
AFAIK, There are no additional costs for using the asynchronous feature of Azure OpenAI.
You can check the cost surge in your Azure OpenAI account, you can use the Cost Analysis tab in the Azure portal.
You can view Azure OpenAI costs in different time intervals and there are filters you can choose to understand the costs.
Please check Plan to manage costs for Azure OpenAI Service for more details.
Incase if that doesn't helps, you can also create a free billing Azure support request.
I hope this helps. Do let me know if you have any further queries.