We show that the scenario of multi-scale Kondo breakdown quantum critical
point (QCP) gives rise to a divergent Gr\"uneisen ratio with an anomalous
exponent 0.7. In particular, we fit the experimental data of
YbRh2(Si0.95Ge0.05)2 for specific heat, thermal expansion, and
Gr\"uneisen ratio based on our simple analytic expressions. A reasonable
agreement between the experiment and theory is found for the temperature range
between 0.4 K and 10 K. We discuss how the Gr\"uneisen ratio is a key
measurement to discriminate between the Kondo breakdown and spin-density wave
theories