Of course we are assuming an effective signal is being received.
I live near Idaho Falls, Idaho with a field elev of 5300ft. If I shoot an approach will GPS be more or less accurate then landing at Seattle or LA?
Does the accuracy of GPS change with aircraft altitude or attitude?
Altitude has little effect, as long as you don't descent into a valley. Mountains shield the GPS signal, resulting generally in lower accuracy and lower integrity levels.
Attitude does matter, steep turns can result in the airframe shielding GPS signals from the antenna. Again, the effect is a lower accuracy and integrity.
The issue is what kind of accuracy you are referring to.
If lateral accuracy, the best geometry is to have satellites distant and just above the horizon. In general, this offers the best solutions for lateral positioning.
If altitude accuracy, it is helpful to have satellites more above you, or at least not near the horizon.
Attitude of the aircraft will impact the reception from one or more satellites, and can cause a change in signal reception which will have the GPS seeking different satellites for it's solution, which might result in short term changes in accuracy.
Furthermore, high altitudes provide more satellites with stronger signals, less terrain masking of distant satellites, and less error inducing signals like multi-path. Lower altitudes tend to have not only more multi-path and similar interference but also terrain interference, where flying near or below terrain will interfere with reception, particularly of distant near horizon satellites. In that case there may be more positional "errors" or less accuracy.
With normal aircraft installations, unshielded by terrain, and with normal constellation visibility, there is little practical difference in normal flight operations.
This answer is a general, aviation oriented answer. There are literally hundreds of peer reviewed papers on this topic, rich with substantiation of various phenomena, models and theories.
Addendum. To address your specific question regarding approach accuracy at higher elevations, it is important to factor in terrain. If one was flying an approach in a valley, there may be some terrain masking of GPS signals. This would be a factor of the relative terrain. However practically speaking, there is no difference. Because the GPS unit will assure that it has adequate signals from adequate satellite transmitters, to assure the precision needed for the approach.
