I am talking about frontier A* where you discard all nodes that have already been visited. This requires the heuristic to be consistent. As opposed to the breadth-first heuristic search which revisits nodes.
A nitpick: "breadth-first heuristic search" is an algorithm developed by Zhou and Hansen and, while it's related to A* (in that it uses an admissible heuristic to prune the search space), it's not actually a variant of A*. (It's not a best-first search algorithm, since it doesn't expand nodes in increasing order of f cost.)