Non-vanishing Temperature can break conformal symmetry (Can anyone show this point explicitly), my question is that in AdS/CFT the temperature of boundary field theory is non-zero, why the boundary field theory whose conformal symmetry is breaking is still a conformal field theory?
AdS/CFT correspondence tells you (among other things) the dual geometry for each state of the boundary theory. As @Arnold says (check the link in the comment), a finite temperature state ("spontaneously") breaks Lorentz, and hence, conformal invariance. That's okay because excited states could break many symmetries of the theory (eg: p-shell of electron cloud around a Hydrogen atom breaks SO(3) symmetry). So it's not like you're losing the conformal symmetry describing the dynamics.
(In fact, I wonder whether one might be able to use something like the ideas behind spurion analysis -- on the finite temperature state, since the temperature is the only thing "breaking" the conformal symmetry, spontaneously :-?)