  Causal diamonds are known to have thermal behavior that can be probed by
finite-lifetime observers equipped with energy-scaled detectors. This
thermality can be attributed to the time evolution of observers within the
causal diamond, governed by one of the conformal quantum mechanics (CQM)
symmetry generators: the noncompact hyperbolic operator $S$. In this paper, we
show that the unbounded nature of $S$ endows it with a quantum instability,
which is a generalization of a similar property exhibited by the inverted
harmonic oscillator potential. Our analysis is semiclassical, including a
detailed phase-space study of the classical dynamics of $S$ and its dual
operator $R$, and a general semiclassical framework yielding basic instability
and thermality properties that play a crucial role in the quantum behavior of
the theory. For an observer with a finite lifetime $\mathcal{T}$, the detected
temperature
  $T_D = 2 \hbar/(\pi \mathcal{T})$ is associated with a Lyapunov exponent
$\lambda_L = \pi T_D/\hbar$, which is half the upper saturation bound of the
information scrambling rate.
