You can use CustomLog to create a very basic log file which is cheap to update. Eventually you can store it somewhere on NFS to limit i/o to the local drive. If you have sufficient memory and need this statistics for a limited period of time you can put the log in memory.
LogFormat "%t %f %T" my_log
CustomLog /some/nfs/mount/point/logs/access_log my_log
Will give you time of the request , filename and the time to honor the request.
For in-memory logging use a tmpfs mount point:
mount -ttmpfs -odefaults,noatime,size=100M,nr_inodes=5k /some/mount/point