  The mechanism of not diverging Gr\"{u}neisen parameter in the quantum
critical heavy-fermion quasicrystal (QC) Yb$_{15}$Al$_{34}$Au$_{51}$ is
analyzed. We construct the formalism for calculating the specific heat
$C_V(T)$, the thermal-expansion coefficient $\alpha(T)$, and the Gr\"{u}neisen
parameter $\Gamma(T)$ near the quantum critical point of the Yb valence
transition. By applying the framework to the QC, we calculate $C_V(T)$,
$\alpha(T)$, and $\Gamma(T)$, which explains the measurements. Not diverging
$\Gamma(T)$ is attributed to the robustness of the quantum criticality in the
QC under pressure. The difference in $\Gamma(T)$ at the lowest temperature
between the QC and approximant crystal is shown to reflect the difference in
the volume derivative of characteristic energy scales of the critical
Yb-valence fluctuation and the Kondo temperature. Possible implications of our
theory to future experiments are also discussed.
