Gauss Law in Arbitrary dimensions

Whether by Gauss law, you mean the dependence of the electrostatic (gravitystatic) potential $phi(x)$, created by some source on the distance from the source (located at the origin), then you start from the equation for the Green's function of the Laplace operator $Delta$ for $x in mathbb{R}^{n}$: $$ Delta phi(x) = q delta(x) $$ After the Fourier transform one gets: $$ k^2 phi(k) = q Rightarrowphi(x) = int frac{d^{n} k}{(2 pi)^{n}} frac{q e^{i kx}}{k^2} $$ This integral in dimensions $n geqslant 3$ gives a power law decay for the potential: $$ phi(x) = frac{C q}{|x|^{n-2}} qquad C = const $$ For the $n = 2$ case the Green function gives a logarithm: $$ phi(x) = C q lnfrac{|x|}{|x_0|}qquad C = const $$ And for the $n=1$ case ($theta(x)$ is the Heavyside function): $$ phi(x) = q x theta(x) $$ The static for general distribution of charges follows from the principle of superposition.

The general law is called Stokes' theorem, https://en.wikipedia.org/wiki/Stokes%27_theorem. In three dimentions, Stokes' theorem has two special forms: the Stokes–Cartan theorem, https://en.wikipedia.org/wiki/Kelvin%E2%80%93Stokes_theorem, and Gauss's theorem, https://en.wikipedia.org/wiki/Divergence_theorem.