Answer: Option B is not true.

Explanation:

Option A states that the half‐life $t_{1/2}$ is $\ln 2$ times $\frac{1}{k}$, where $k$ is the decay (rate) constant.

$ t_{1/2} \;=\; \frac{\ln 2}{k} \;\;=\;\ln 2 \times \frac{1}{k}. $

This is correct.

Option B states that the decay constant $k$ increases with an increase in temperature. However, for radioactive (nuclear) decay processes, the decay constant is essentially independent of external factors such as temperature and pressure. Hence, saying that it increases with temperature is not true.

Option C states that the decay constant does not depend upon temperature, which is true for radioactive decay.

Option D states that the amount of radioactive substance left after three half‐lives is $\tfrac{1}{8}$ of the original. Since one half‐life leaves $\tfrac{1}{2}$ of the sample, three half‐lives leave $\left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8}$. This is correct.

Therefore, the statement that is not true is Option B.