We demonstrate the strong impact of the oscillator agility on the line broadening by studying spin transfer induced microwave emission in MgO-based tunnel junctions with current. The linewidth is almost not affected by decreasing the temperature. At very low currents, a strong enhancement of the linewidth at low temperature is attributed to an increase of the non linearity, probably due to the field-like torque. Finally we evidence that the noise is not dominated by thermal fluctuations but rather by the chaotization of the magnetization system induced by the spin transfer torque.