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.