  The mass of the graviton, if nonzero, is usually considered to be very small,
e.g. of the Hubble scale, from several observational constraints. In this
paper, we propose a gravity model where the graviton mass is very small in the
usual weak gravity environments, below all the current graviton mass bounds,
but becomes much larger in the strong gravity regime such as a black hole's
vicinity. For black holes in this model, significant deviations from general
relativity emerge very close to the black hole horizon and alter the black hole
quasi-normal modes, which can be extracted from the ringdown waveform of black
hole binary mergers. Also, the enhancement of the graviton mass near the
horizon can result in echoes in the late time ringdown, which can be verified
in the upcoming gravitational wave observations of higher sensitivity.
