You can init the AStar variable one time (add points, connect points)
Later you just update it (add more points, disconnect points, reconnect points)
To save what is in the AStar for later use, create a dictionary to save the points and thier info. Then, you just save the dictionary into a file using tojson().
To load what you have saved, use parsejson() to get the dictionary you saved. Then, iterating the points and init the AStar base on what info each point contains.
(If you don't know how to save infos into a file: var file = File.new(), file.open(), file.storeline(tojson(Your_Dictionary)) You can look up the class File in godot docs)