  At first glance, thermodynamic properties of gravity with asymptotically AdS
conditions and those with box boundary conditions, where the spatial section of
the boundary is a sphere of finite radius, appear similar. Both exhibit a
similar phase structure and Hawking-Page phase transition. However, when we
introduce a U(1) gauge field to the system, discrepancies in thermodynamic
properties between the two cases has been reported in [7] (JHEP 11 (2016) 041).
In this paper, by accepting the assumption that all Euclidean saddles
contribute to the partition function, I found that these discrepancies are
resolved due to the contribution from the "bag of gold (BG)," which is the
class of Euclidean geometries whose area of bolt is bigger than that of the
boundary. As a result, the Hawking-Page phase structure is restored, with the
unexpected properties that the upper bound of thermodynamic entropy is always
larger than the boundary area divided by 4G when the chemical potential is
non-zero, and that such high entropy states are realized at sufficiently high
temperature.
