  We introduce the quasi-hyperbolicity constant of a metric space, a rough
isometry invariant that measures how a metric space deviates from being Gromov
hyperbolic. This number, for unbounded spaces, lies in the closed interval
$[1,2]$. The quasi-hyperbolicity constant of an unbounded Gromov hyperbolic
space is equal to one. For a CAT$(0)$-space, it is bounded from above by
$\sqrt{2}$. The quasi-hyperbolicity constant of a Banach space that is at least
two dimensional is bounded from below by $\sqrt{2}$, and for a non-trivial
$L_p$-space it is exactly $\max\{2^{1/p},2^{1-1/p}\}$. If $0 < \alpha < 1$ then
the quasi-hyperbolicity constant of the $\alpha$-snowflake of any metric space
is bounded from above by $2^\alpha$. We give an exact calculation in the case
of the $\alpha$-snowflake of the Euclidean real line.
