Early termination of A* with weak heuristic if solution is known
In order to prove that the solution is optimal, with a consistent heuristic A* needs to expand all states with f &lt; f* (the perfect solution cost) and it also may expand some states with f = f* so it seems hard that knowing an upper bound on the number of expansions can help to reduce the number of node expansions without modifying the heuristic. 

A potential way of helping A* in this context may be that of heuristic selection. That is, in cases where more than one heuristic is available, meta-reason about what subset of heuristics should be used [1,2].

In this scenario it could be possible to use the sub-optimal path to select which ones to use (e.g., it may be worth it to compute more expensive heuristics because they have greater chances of pruning the node). However, this is an open problem and I'm not aware of any work on that direction.  

[1] David Tolpin, Tal Beja, Solomon Eyal Shimony, Ariel Felner, Erez Karpas:
Toward Rational Deployment of Multiple Heuristics in A. IJCAI 2013: 674-680

[2] Levi H. S. Lelis, Santiago Franco, Marvin Abisrror, Mike Barley, Sandra Zilles, Robert C. Holte:
Heuristic Subset Selection in Classical Planning. IJCAI 2016: 3185-3191