ActivityTimeline is Atlassian Marketplace application and is paid via Atlassian on the same invoice with all Atlassian products.
For Cloud based version of ActivityTimeline app billing is done through Atlassian Marketplace on the same invoice that your Jira is being billed. Depending on your Jira subscription model you will be charged either monthly or annually.
For more details on the licensing and purchasing we recommend you checking the following page: https://www.atlassian.com/licensing/marketplace#licensing-purchasing
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Once you purchased a license thorught he through the Marketplace page, you will receive a license key, that you need to insert in Jira Settings ->Manage Apps->Edit → Manage Apps → Edit your ActivityTimeline License key:
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For more information regaring regarding this topic we recommend checking the following page: Customer Q & A - Licensing and Purchase
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