Is 'double max-$k$ vertex-cover' NP-hard?
It was easier for me to show that the $k$-clique problem is a special case of this problem.

On an input graph $G$ and an integer $k$, the this special case is defined by  $G$,$k$,$t$, with $t=\binom{k}{2}=k(k-1)/2$.
Observe that some $V'\subseteq V$ of size $k$ is a clique in $G$ if and only if the set $V'$ covers at least $\binom{k}{2}$ edges twice. 

Since the NP-hard problem $k$-clique is a special case of 'double max-$k$ vertex-cover', the problem 'double max-$k$ vertex-cover' is NP-hard.