  In Yukawa cosmology, a recent discovery revealed a relationship between
baryonic matter and the dark sector. The relation is described by the parameter
{\alpha} and the long-range interaction parameter {\lambda} - an intrinsic
property of the graviton. Applying the uncertainty relation to the graviton
raises a compelling question: Is there a quantum mechanical limit to the
measurement precision of the Hubble constant (H0)? We argue that the
uncertainty relation for the graviton wavelength {\lambda} can be used to
explain a running of H0 with redshift. We show that the uncertainty in time has
an inverse correlation with the value of the Hubble constant. That means that
the measurement of the Hubble constant is intrinsically linked to length scales
(redshift) and is connected to the uncertainty in time. On cosmological scales,
we found that the uncertainty in time is related to the look-back time
quantity. For measurements with a high redshift value, there is more
uncertainty in time, which leads to a smaller value for the Hubble constant.
Conversely, there is less uncertainty in time for local measurements with a
smaller redshift value, resulting in a higher value for the Hubble constant.
Therefore, due to the uncertainty relation, the Hubble tension is believed to
arise from fundamental limitations inherent in cosmological measurements.
Finally, our findings indicate that the mass of the graviton fluctuates with
specific scales, suggesting a possible mass-varying mechanism for the graviton.
