For spatially bounded free fields, the Bekenstein bound states that the specific entropy satisfies the inequality SE≤2πR, where R stands for the radius of the smallest sphere that circumscribes the system. The validity of the Bekenstein bound in the asymptotically free side of the Euclidean (λφ4)d scalar field theory is investigated. We consider the system in thermal equilibrium with a reservoir at temperature β-1 and defined in a compact spatial region without boundaries. Using the effective potential, we discuss the thermodynamic of the model. For low and high temperatures the system presents a condensate. We present the renormalized mean energy E and entropy S for the system and show in which situations the specific entropy satisfies the quantum bound. © 2010 The American Physical Society.