  The extended minimal geometric deformation (EMGD) is employed on the fluid
membrane paradigm, to describe compact stellar objects as Bose--Einstein
condensates (BEC) consisting of gravitons. The black hole quantum portrait,
besides deriving a preciser phenomenological bound for the fluid brane tension,
is then scrutinized from the point of view of the configurational entropy. It
yields a range for the critical density of the EMGD BEC, whose configurational
entropy has global minima suggesting the configurational stability of the EMGD
BEC.
