Kubernetes | Multiple pods - performance problem
For cache issues have you considered :
Ingress controllers like nginx to cache static content and deliver from it straight? https://medium.com/@vdboor/using-nginx-ingress-as-a-static-cache-91bc27be04a1
CDN may be if the cache is not private or dynamic in nature ?
With increasing pods, increasing times to me doesn't sound like its a cache issue or not the cache alone. The webserver is playing a big part or the load balancer an/or the firewall sitting in the front is capping the bandwidth. Round trip from browser to pod back to browser should be same if you have 1 or 100 pods provided there is no network latency. In your case an increase traffic is slowing the connection speed. I have had similar issues with the network capping bandwidth in front of the pods.