Hi,
Unfortunately you cannot stop/start Azure Bastion to save costs. You are charged per hour for Bastion from the time you create it until you delete it.
Excerpt from pricing page:
2 Azure Bastion is billed hourly from the moment the resource is deployed until the resource is deleted, regardless of outbound data usage. The hourly pricing will be based on the SKU selected, number of scale units configured, and data transfer rates.
Azure Bastion pricing
https://azure.microsoft.com/en-us/pricing/details/azure-bastion/
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Thanks.
-TP