  A universal relation between the leading correction to the entropy and
extremality was gotten in the work of Goon and Penco. In this paper, we extend
this work to the massive gravity and investigate thermodynamic extremality
relations in a topologically higher-dimensional black hole. A rescaled
cosmological constant is added to the action of the massive gravity as a
perturbative correction. This correction modifies the extremality bound of the
black hole and leads to the shifts of the mass, entropy, etc. The Goon-Penco
relation is gotten. Regarding the cosmological constant as a variable related
to pressure, we get the thermodynamic extremality relations between the mass
and pressure, charge, parameters $c_i$ by accurate calculations, respectively.
Finally, these relations are verified by a triple product identity, which shows
that the universal relation exists in black holes.
