Transistor varies with Temperature
If you look at the Ebers -Moll model, one can see that:

$$I_C = \dfrac{\beta_F}{\beta_F +1}I_E$$

which can be rewritten as

$$\beta_F +1 = \beta_F \dfrac{I_E}{I_C}$$

$$1+\dfrac{1}{\beta_F}=\dfrac{I_E}{I_C}$$

$$ \beta_F = \dfrac{1}{\frac{I_E}{I_C}-1}$$

Where 

$$I_E = I_{ES}\{e^{\frac{V_{BE}}{V_T}}-1\}$$

where $$V_T=\dfrac{k_BT}{q}$$ is the thermal voltage.

Here it should be clear that the temperature cause a change in the thermal voltage which leads to a change in beta.

The equation is basically a statement about the distribution of energy per unit of charge, as the temperature increases, the exponential in the emitter current equation decreases, which causes the ratio of collector current to emitter current to increase, and since that ratio is in the denominator of the equation for beta, it has an inverse relationship, which causes beta to increase. So as temperature increases, beta increases.