The expectation over $x$ of a covariance between variables $A$ and $B$ (where the distribution of $B$ varies according to $x$) is equal to the covariance of $A$ with the expectation of $B$ over $x$:
$$ E_x[Cov(A,B)]=Cov(A,E_x[B]) $$
I wonder, then, whether it is also true that the variance of the covariance follows the same pattern? i.e.
$$ Var_x[Cov(A,B)]=Cov(A,Var_x[B]) $$
This seems like it might be true: can anyone confirm this?