I also encounter this issue.I would like to provide some of my thought.
My team would need to support some APIs with heavy operation and it would be open to unauthenticated users which is design by business logic.
That's why we need to restrict api requests to our app clients.
The API call is stateless and irrelative with caching and proxies.
In the other hand, some malicious attack like CSRF, you should also provide some additional protection on you API to prevent request sending from untrusted way.
There are several mechanism we considered.
Using HTTP header
This is untrusted and very easy to crack.
Use one static random generated API Key
Very common and easy-implementation way. Server generated one static random string as key and client must carry when sending request.
If you have to support web, this would be leak by web console.But if you only support app client and restrict your API connection with HTTPs. This should be safe enough.
Dynamic change API key with AES crypto algorithm
To prevent MITM or static API key is leak, I proposed to use AES crypto algorithm and encrypt current timestamp.
When server receive, decrypt and check whether the request is valid or not.
You can also append some string as salt to make the mechanism harder to brute force attack.
You can do as much effort to make it harder to crack, but it would never be absolutely 100% safe.
Hackers can still reverse engineer your app to see how the encryption works.
All you can do is making it harder.
This is my propose and hope it could inspire you.
If you have any other better solutions or find some bug in my proposal, please let me know.