  Different theories of gravity can admit the same black hole solution, but the
parameters usually have different physical interpretations. In this work we
study in depth the linear term $\beta r$ in the redshift function of black
holes, which arises in conformal gravity, de Rham-Gabadadze-Tolley (dRGT)
massive gravity, $f(R)$ gravity (as approximate solution) and general
relativity. Geometrically we quantify the parameter $\beta$ in terms of the
curvature invariants. Astrophysically we found that $\beta$ can be expressed in
terms of the cosmological constant, the photon orbit radius and the innermost
stable circular orbit (ISCO) radius. The metric degeneracy can be broken once
black hole thermodynamics is taken into account. Notably, we show that under
Hawking evaporation, different physical theories with the same black hole
solution (at the level of the metric) can lead to black hole remnants with
different values of their physical masses with direct consequences on their
viability as dark matter candidates. In particular, the mass of the graviton in
massive gravity can be expressed in terms of the cosmological constant and of
the formation epoch of the remnant. Furthermore the upper bound of remnant mass
can be estimated to be around $0.5 \times 10^{27}$ kg.
