  Gravitational thermodynamics and gravitoscalar thermodynamics with $S^2
\times \mathbb{R}$ boundary geometry are investigated through the partition
function, assuming that all Euclidean saddle point geometries contribute to the
path integral and dominant ones are in the $B^3 \times S^1$ or $S^2 \times
Disc$ topology sector. In the first part, I concentrate on the purely
gravitational case with or without a cosmological constant and show there
exists a new type of saddle point geometry, which I call the "bag of gold(BG)
instanton," only for the $\Lambda>0$ case. Because of this existence,
thermodynamical stability of the system and the entropy bound are absent for
$\Lambda>0$, these being universal properties for $\Lambda \leq 0$. In the
second part, I investigate the thermodynamical properties of a gravity-scalar
system with a $\varphi^2$ potential. I show that when $\Lambda \leq 0$ and the
boundary value of scalar field $J_{\varphi}$ is below some value, then the
entropy bound and thermodynamical stability do exist. When either condition on
the parameters does not hold, however, thermodynamical stability is (partially)
broken. The properties of the system and the relation between BG instantons and
the breakdown are discussed in detail.

General Relativity and Quantum Cosmology;High Energy Physics - Theory