Does Kepler's 3rd law of planetary motion violate the first postulate? Consider a distant observer traveling at .866 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. According to his/her wristwatch the observer measures the earth's orbital period around the sun to be 730.5 days, correct?
But the observer also measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.
So it appears as if Kepler's 3rd law of planetary motion is only valid in the rest frame of the solar system.  Does this violate the first postulate of special relativity?
If so then how can Kepler's 3rd law be made frame-invariant?
 A: 
It appears as if Kepler's 3rd Law of Planetary Motion is only valid in the rest frame of the solar system. Does this violate the First Postulate of Special Relativity?

No. It just means that Kepler’s laws are not actually laws of physics. Instead, they are approximations to the laws of physics in the non-relativistic limit
A: I think the analysis being performed in this Q and A is based on a simple misunderstanding of relativity.  The Principle of Relativity (referred to above as the 1st Postulate) states that the laws of physics are valid in any inertial frame of reference locally in that frame.  This means that if the observer moving at $0.866c$ (w.r.t the Solar System) does an experiment in a laboratory on his own spacecraft, where everything in the laboratory is stationary with respect to him, all normal physical laws of motion will hold good.  Every result in his lab will agree with the results in our labs on Earth.  If his entire solar system around him is speeding through the galaxy at $0.866c$ (e.g. like this), Kepler's 3rd Law will hold for that system just as well as for ours.
The Principle of Relativity does not say that far away events and objects will look the same regardless of your state of motion.  In fact, the groundbreaking aspect of the theory of relativity was that, in order for the first paragraph to be true, events and objects moving relative to the viewer must look different.  For example, in Einstein's original 1905 paper, he shows that a rigid sphere (Sec. 4), seen from a stationary point of view, will be an ellipsoid compressed in the direction of motion when seen from a moving frame.  This will apply to the shape of the planetary orbits in the original example (especially if the orbital speeds of the planets are much less than the speed of light, as is the case), as well as the shapes of the planets themselves.  Likewise, light which is a certain frequency when viewed in the same frame as the source will be blue-shifted and higher-intensity when the source is moving toward the observer (Sec. 7).
All of these effects stem from an observer viewing an object or event that is far away and/or moving relative to him.  But Special Relativity guarantees that for events in your immediate vicinity and stationary with respect to you, the laws of physics do not depend on your rate of motion (with respect to something else).  Thus, there is no preferred frame of "absolute rest," because everything is at absolute rest with its immediate environment.
A: The first answer seeks to invalidate Kepler’s laws as only “approximations to the [actual] laws of physics in the non-relativistic limit”.  It is then left to the reader’s imagination as to what those relativistically correct laws of physics might look like in this case.  This seems too dismissive.
I will demonstrate that it is in fact Kepler’s third law which invalidates special and/or general relativity, and not the other way round.  The method of reductio ad absurdum will be employed in the demonstration.
Consider the situation described above in the original question.  The distant observer’s trajectory guarantees that the earth’s orbit around the sun remains un-distorted by Lorentz contraction.   The distant observer watches Big Ben in London with a powerful telescope. Allowing for relativistic doppler, at .867c the distant observer measures Big Ben to be keeping time at only half the rate of his own proper time wristwatch.  This is in accordance with special relativity.
From the standpoint of the distant observer, in order for Big Ben and the earth’s orbital period to remain synchronized at 730.5 revolutions of Big Ben’s little hand for every revolution that the earth makes around the sun, the earth’s orbital velocity will have to slow down to half the speed that we measure in the solar system’s rest frame.  So far so good.
But a problem arises when we consider that while the earth’s orbital velocity has been cut in half, the spacetime curvature in which the earth is traveling has not been reduced at all.  That remains invariant for all observers in all inertial frames:
Does the spacetime curvature in the vicinity of a massive body increase, decrease or remain unchanged with the increasing velocity of an observer?
We have constructed a situation in which the earth is traveling in the identical spacetime curvature, but at a velocity we know to be far too slow to sustain its orbit around the sun.  In this example we find to our embarrassment that the fate of the earth rests in the hands of a distant observer who controls whether or not the earth will spiral into the sun.
A: While Kepler's third law is known to be true in the rest frame of the solar system, it can be demonstrated to be invalid when applied to the solar system by observers in other inertial frames of reference. This conflicts with special relativity's first postulate which requires that the laws of physics be valid and take the same form in all inertial frames of reference.
To a first approximation Kepler's third law of planetary motion states
$$\frac{R^3}{T^2}=K$$
where $K$ has the numerical value $7.5x10^{-6}$ when the orbital period $T$ is measured in $days$ and the semi-major axis of the orbit $R$ is measured in astronomical units $AU$.
The earth's orbital period in the solar system's rest frame is
$T=365\,days$
The Lorentz time dilation for the earth's orbital period between the solar system's rest frame and all other inertial frames is
$T'=\biggl(\frac{365\,days}{\gamma}\biggr)$
The semi-major axis (and the semi-minor axis) of the earth's orbit is identical for all inertial frames where the observer's velocity vector is collinear with the sun's axis of rotation
$R'=R=1AU$
So the relativistic expression of Kepler's third law when the observer's velocity vector is collinear with the sun's axis of rotation is
$$\frac{1AU^3}{\biggl(\frac{365\,days}{\gamma}\biggr)^2}=7.5\times10^{-6}\frac{AU^3}{days^2}$$
From this if follows that
$133333\,days^2 =\biggl(\frac{365\,days}{\gamma}\biggr)^2$
$365\,days = \frac{365\,days}{\gamma}$
$365\,days-365\,days\sqrt{1-\frac{v^2}{c^2}}=0$
The last equation is only valid in the rest frame of the solar system where $v=0$.  The equation is invalid in all other inertial frames which is a violation of the first postulate of special relativity.
