Do Black Strings Violate Black Hole Uniqueness in Higher Dimensions? Most black string solutions are obtained by adding flat directions to lower dimensional black holes, for instance in the Gregory-Laflamme instability argument. I had the following question;

Is this a violation of black hole uniqueness in higher dimension since
  the asymptotics of these black strings/black hole solutions are
  different?

 A: “Black hole uniqueness” is not a single statement but a variety of results with different prerequisites: static vs. stationary, vacuum vs. Einstein–Maxwell vs. other matter fields; asymptotic flatness, cosmological constant, analyticity etc. So whether black strings violate uniqueness in higher dimensions depends on what specific theorem is under consideration.
For example, there is uniqueness theorem  for asymptotically flat static vacuum black hole solutions in higher dimensional space-times (gr-qc/0203004): all black holes satisfying its requirements are from Schwarzschild–Tangherlini family (a higher dimensional generalization of Schwarzschild metric). Static black strings could not serve as counterexamples for precisely the reason mentioned by OP: they are not asymptotically flat.
On the other hand, if we are looking at stationary, vacuum, asymptotically-flat solutions then the following solutions are known in closed form:


*

*the  Myers–Perry black holes (generalizations of the Kerr metric with spherical horizon topology); 

*Emparan–Reall  black  rings  with $_2×_1$ horizon  topology; 

*Generalization of black ring by Pomeransky–Senkov to include a second angular-momentum parameter;

*the “Black Saturn” solutions  discovered by Elvang and Figueras (essentially a black ring around black hole with spherical horizon topology). 


Black rings could be seen as loops of black string and these solutions indeed violate the higher dimensional generalization of uniqueness for asymptotically flat, stationary, vacuum black hole, so for this version of “uniqueness” the answer to the titular question is positive.
More on higher dimensional black holes could be found in this Living Review.
