Why can β not be linearly proportional to T, that is β=constant×T?

Why can $\beta$ not be linearly proportional to T, that is $\beta =constant×T$
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salumeqi
Because by convention, we want to write $\beta$ as the coefficient in front of energy E in the exponent $\mathrm{exp}\left(-\beta E\right)$. Exponents have to be dimensionless so $\beta$ has to have units of inverse energy. That's why it has to be objects such as $\beta =1/kT$ because $kT$ has units of energy. The latter statement holds because the energy per degree of freedom increases with the temperature. At the end, that's the fundamental answer to your question. The exponential is $\mathrm{exp}\left(-E/kT\right)$ because $E\sim kT$ and "hot" means "highly energetic". It's linked to the fact that the temperature has an absolute lower bound, the absolute zero, much like energy is bounded from below.