Hot spaces with positive cosmological constant in the canonical ensemble: de Sitter solution, Schwarzschild-de Sitter black hole, and Nariai universe

In a space with positive cosmological constant $\Lambda$, we consider a black hole surrounded by a heat reservoir at radius $R$ and temperature $T$, i.e., we analyze the Schwarzschild-de Sitter black hole in a cavity. We use the Euclidean path integral approach to quantum gravity to study its canonical ensemble and thermodynamics. We give the action, energy, entropy, temperature, and heat capacity. $T$, $\Lambda$, the black hole radius $r_+$, and the cosmological horizon radius $r_{\rm c}$, are gauged in $R$ units to $RT$, $\Lambda R^2$, $\frac{r_+}{R}$, and $\frac{r_{\rm c}}{R}$. The whole extension of $\Lambda R^2$, $0\leq\Lambda R^2\leq 3$, is divided into three ranges. The first, $0\leq\Lambda R^2<1$, includes York's Schwarzschild black holes. The second range, $\Lambda R^2=1$, opens up a folder of Nariai universes. The third range, $1<\Lambda R^2\leq 3$, is unusual. One feature here is that it interchanges the cosmological horizon with the black hole horizon. The end point, $\Lambda R^2=3$, only existing for infinite $RT$, is a cavity filled with de Sitter space, except for a singularity, with the cosmological horizon coinciding with the reservoir. For the three ranges, for low temperatures, there are no black holes and no Nariai universes, the space is hot de Sitter. The value of $RT$ that divides the nonexistence from existence of black holes or Nariai universes, depends on $\Lambda R^2$. For each $\Lambda R^2\neq1$, for high temperatures, there is one small and thermodynamically unstable black hole, and one large and stable. For $\Lambda R^2=1$, for high temperatures, there is the unstable black hole, and the neutrally stable Nariai universe. Phase transitions can be analyzed. The transitions are between the black hole and hot de Sitter and between Nariai and hot de Sitter. The Buchdahl radius, the radius for collapse, plays an interesting role in the analysis.