AAM: how to cache home page? (when path = /)

I think what you are seeing is what is in SOL10569: BIG-IP WebAccelerator X-PvInfo response headers may demonstrate inconsistent object cache promotion sequence . They changed the header name in AAM but the sequence should still be the same.

The expected object promotion sequence through the different caches is as follows:

• The initial request for a cacheable object received by the BIG-IP LTM is passed to the BIG-IP WebAccelerator and proxied to the origin server. The response from the origin server is served to the client with an S10201 X-PvInfo header code and cached to the BIG-IP WebAccelerator cache, or Smart Cache disk.
• When the next request for the object is received by the BIG-IP LTM and passed to the BIG-IP WebAccelerator, the object is served to the client from Smart Cache with an S10102 X-PvInfo header code and promoted to memory Smart Cache.
• When the next request for the object is received by the BIG-IP LTM and passed to the BIG-IP WebAccelerator, the object is served to the client from memory Smart Cache with an S10101 X-PvInfo header code and promoted to the BIG-IP LTM RAM Cache.
• Subsequent requests for the object received by the BIG-IP LTM are served directly from the BIG-IP LTM RAM Cache with an S11101 X-PvInfo header code and not passed to the BIG-IP WebAccelerator until the RAM Cache maximum age is reached.