The Schwarzschild–de Sitter solution in the Einstein theory with a positive cosmological constant $$\Lambda =m^2/\alpha $$ becomes an exact solution to the de Rham–Gabadadze–Tolley (dRGT) nonlinear massive gravity theory with the mass parameter $$m$$ when the theory parameters $$\alpha $$ and $$\beta $$ satisfy the relation $$\beta =\alpha ^2$$. We study the perturbative behavior of this black hole solution in the nonlinear dRGT theory with $$\beta =\alpha ^2$$. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned, in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the Stückelberg field sector at the linear perturbation level in this background. The physical significance of this degeneracy depends on how the Stückelberg fields couple observable fields.

We calculate explicitly the black-hole temperature for the Schwarzschild de-Sitter solution inside massive gravity by defining the Killing-vector in the direction of the Stückelberg function. We then consider the conditions which an observer in massive gravity has to obey in order to agree with the standard results of General Relativity.

I derive the Komar mass/function for the Schwarszchild\char21{}de Sitter black hole inside the de Rham\char21{}Gabadadze\char21{}Tolley nonlinear theory of massive gravity by taking the usual notion of timelike Killing vector in unitary gauge. The de Rham\char21{}Gabadadze\char21{}Tolley Komar function depends on the dynamics of the St\"uckelberg fields through the gauge transformation function. It goes to the standard value obtained in general relativity if the spatial derivative of the gauge function vanishes. In such a case, the (gauge) function corresponds to the usual notion of time as in general relativity.