Many more experiments in dark energy need to be done, and some are, as listed below.
In case you don't know what dark energy is, its the "stuff" that makes up about 70% of the universe we live in. Its not matter like you and I, its energy. I don't know exactly what energy is either, the best answer I heard years ago in my Conceptual Physics class was "its' the stuff that make stuff do stuff". Anyways a big question in physics is: What is this dark energy?
The idea of dark energy started from results on the measurements of certain type of supernovas that show that space wasn't just expanding it was that this expansion is increasing. The Nobel Prize in Physics for 2011 was given for this experimental work. This means that the expansion of the universe is accelerating. For some reason the rate of expansion of empty space was increasing. The answer to this riddle is that there is some kind of energy in all of space causing this acceleration of the universe. This has been called dark energy. I know I'm not saying this correctly, but hopefully you get the idea. Answers to many questions about dark energy can be obtained from Sean Carroll's blog here. There are still more questions. It's interesting that one thing that is being studied is an equation of state for the dark energy. Equations of state show up in all different areas of physics. I remember folks talking about them in nuclear physics 30 years ago or so.
The real neat thing is that many more measurements about dark energy need to be done, So there are a variety of questions. From all the sources I've read it is assumed that the dark energy is constant throughout the universe. This needs to be tested and fortunately some experiments are being done.
One experimental test I've come across is the HETDEX group described here.
Another experiment is the Dark Energy survey here This site is really good and has lots of understandable information about dark energy searches they will be doing. They will be taking lots of data!!
I asked the following question at stack exchange and here's some answers.
So the answers I have gotten so far are that dark energy might be anistropic but not by much. That's the understanding I also get from reading the paper in arxiv given in the answer. So more data are needed!
Now what about dark energy as a function of the distance from us? I can't locate any measurements of this as such.
The problem seems to be in getting a handle on how to measure the dark energy and what is it exactly? Since dark energy is related to gravity is the particle for dark energy a graviton? Probably not since the gravition is the field particle for gravity. Gravitons are purely theoretical and no direct gravitons have been observed in a manner similar to experiments that have single photons being observed in the lab all the time. No one really knows what is dark energy. Is it whats left over from the inflaton field?
To answer what exactly is dark energy and to figure out and perform an experiment to measure it directly is quite a challenge. I suspect a meeting with the King of Sweden would happen if you can answer these questions.
From reading about dark matter searches going on with a variety of land and space based observatories maybe they can say something about dark energy also. A good place to look for recent dark matter searches is from a recent Dark Matter meeting at UCLA here. There are some good talks to examine. Again lots of neat experiments for the future!
An exciting time for physics and astronomy!!!