My understanding was that thinking still gets encrypted, shared with clients, and reingested by Anthropic with each new prompt [1]. Which means it would cost more than normal tokens, since it has to be decrypted/encrypted with every transaction.
[1] https://blog.cryptographyengineering.com/2026/05/29/fooling-...
Edit: other comments under this post seem to indicate that thinking tokens are cached on the server side as well? I'm a bit confused.