Variance in Fst in the infinite island model
From Lewontin and Krakauer 1973, the ratio

$$\frac{F_{ST}(d-1)}{\bar F_{ST}}$$

approximatively follows $\chi^2$ distribution of degree $k=d-1$. Here $d$ is the number of demes (number of islands), $F_{ST}$ is the random variable of the $\chi^2$ distribution and $\bar F_{ST}$ is the average $F_{ST}$ that is $\bar F_{ST} = \frac{\sum F_{ST}}{n}$, where $n$ is the number of $F_{ST}$ values.

The variance of a $\chi^2$ distribution is $2k$, therefore

$$var\left(\frac{F_{ST}(d-1)}{\bar F_{ST}}\right) = 2d-2$$

Taking $\frac{d-1}{\bar F_{ST}}$ out of the ratio, the variance of $F_{ST}$ becomes 

$$var(F_{ST})=\left(\frac{d-1}{\bar F_{ST}}\right)^2(2d-2)$$

, which simplifies into

$$var(F_{ST}) = \frac{2(d-1)^3}{\bar F_{ST}^2}$$



The above expression is probably the most interesting result but one could go further and express the variance independently from the mean (by replacing $\bar F_{ST}$ by Slatkin 1991 expectation for $\bar F_{ST}$ in a finite island. It yields to

$$var(F_{ST}) = \frac{2(d-1)^3}{\left(\frac{1}{1+4Nm(\frac{d}{d-1})^2}\right)^2}$$

, which again "simplifies" into

$$var(F_{ST}) = \frac{2 \left(4 d^2 m N+d^2-2 d+1\right)^2}{d-1}$$