  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.
