What class of spacetimes has an associated temperature? It is very well known that the Schwarzschild metric carries a temperature inversely proportional to the mass. Is there a much wider class of spacetimes that has temperatures associated with it? What properties a spacetime must have to carry temperature?
 A: The "temperature" of the Schwarzschild metric is not exactly a temperature. If Hawking radiation is exactly thermal then the unitarity of quantum mechanics will be violated. There is growing consensus that quantum gravity is unitary and information is conserved. To preserve unitarity the radiation emitted cannot be uncorrelated as you would expect for thermal radiation.
If you meant to ask what class of black holes will radiate energy outside their event horizons, then the answer would be all black holes. The usual proof of Hawking radiation using Bogoliubov transformations assumes that space-time is “eternal” and that it is stationary in the far future and in the far past. Symmetrical black holes are used for proving because they make it easier to calculate.
But even the most general black holes which do not have symmetries should radiate energy because using the principle of equivalence and using the Rindler effect we can expect Hawking radiation. In the unsymmetrical cases, we will not have a unique "temperature". When a black hole is in equilibrium its surface gravity $\kappa$ will be uniform on the event horizon. If $\kappa$ is uniform then Hawking "temperature" is $\frac{\kappa}{2\pi}$.
A: As commenters already suggested, a Killing horizon is necessary to associate a temperature to a spacetime. This also means that spacetime needs timelike Killing vector field (KVF), then we can define a Hartle–Hawking state on this spacetime background regular across this horizon so that observers static with respect to that KVF would register a thermal bath. But an additional ingredient needed is that there must be a “natural” norm to that KVF, otherwise we have different
observers for example at points $p_{1,2}$ who would register different temperatures $T_{1,2}$ connected through Tolman's relation:
$$
T_1 \sqrt{−g_{00}(p_1)}=T_2 \sqrt{−g_{00}(p_2)}, 
$$
where the time coordinate corresponds to this timelike KVF.
But without a natural reference point which would correspond to unital norm of KVF there is no way for different observers to calibrate their observations and agree on the true temperature of spacetime, we would just have a bunch of temperatures for individual observers.
Example 1: Schwarzschild spacetime. There is a unique timelike KVF ($\partial_t$, where $t$ is the Schwarzschild time) and a natural way to assign norm to this field: by declaring it as time of asymptotic observer at rest with the black hole. This means that $T=\frac{\hbar c^3}{8 \pi k_\text{B} G M}$ is the true temperature of spacetime and could be measured as a temperature of Hawking radiation coming from the black hole by an observer far away from it. Observers held static close to that black hole would measure a different temperature but could introduce a blueshift factor in an unambiguous way. Note, that even when there are no points of spacetime that would register the exact spacetime temperature, the existence of reasonable asymptotics still allows us to associate a unique norm to KVFs.
Example 2: Rindler spacetime. This is a Minkowski spacetime as seen by uniformly accelerated observers. Such an observer would measure a finite Unruh temperature $T=\frac{\hbar a}{2\pi k_\text{B} c}$ (where $a$ is the  acceleration), that  could be attributed to the Rindler horizon. The timelike KVF associated with this spacetime is a boost generator along a particular direction with a specific point as a center. There are a lot of KVFs in Minkowski spacetime, so there are Rindler horizons through every point and with all orientations. But we can make a  KVF of a specific boost unique by introducing matter fields that commute just with this specific KVF and don't commute with the rest of possible KVFs. Still, even in this case there is no natural norm to this KVF since rescaling it would introduce a new set of reference observers no better and no worse than the initial set. So we have only observer temperatures for this spacetimes that go arbitrarily high and arbitrarily low, with no specific temperature associated  to this spacetime.
Example 3: Schwarzschild–de Sitter spacetime.
There is a timelike KVF here and well behaved asymptotics, which allow us to normalize this KVF. But, the spacetime has two Killing horizons: the event horizon of a black hole at the center and a cosmic horizon and those horizons have different temperatures. So, there is no temperature of a spacetime, but instead two temperatures of two horizons. And it is impossible to define a Hartle–Hawking state that would be regular across both horizons.
So, we obtain the following requirements for a well defined temperature of a spacetime:

*

*Unique timelike Killing vector field that allows its unambiguous normalization.


*(A single) Killing horizon associated with this KVF.
