R-Tree multipath search - remember the path?
Sure, you can cache the results of lookups into a R-tree (or any other data structure).  Simply build a cache (e.g., a hashtable) where after each lookup into the R-tree, you update the cache; and before doing a lookup into the R-tree, check the cache first.  You don't need to store the entire path in the cache: it's enough to just store the results of the search.

Whether this will improve performance or not will depend on your particular workload.