  It is known that a solution of remnant were suggested for black hole ground
state after surface gravity is corrected by loop quantum effect. On the other
hand, a Schwarzschild black hole in asymptotic Anti-de Sitter space would
tunnel into the thermal soliton solution known as the Hawking-Page phase
transition. In this letter, we investigate the low temperature phase of
three-dimensional BTZ black hole and four-dimensional AdS Schwarzschild black
hole. We find that the thermal soliton is energetically favored than the
remnant solution at low temperature in three dimensions, while Planck-size
remnant is still possible in four dimensions. Though the BTZ remnant seems
energetically disfavored, we argue that it is still possible to be found in the
overcooled phase if strings were present and its implication is discussed.
