Speed of response increases with gain

Speed of response is measured by the time constant (first order) or natural frequency (second order); both improve as the loop gain $K$ rises.

First-order system

With unity feedback around $G(s) = \dfrac{K}{\tau s+1}$:

$$\frac{C(s)}{R(s)} = \frac{K}{\tau s+1+K} = \frac{K/(1+K)}{\dfrac{\tau}{1+K}s+1}$$ $$\tau_{cl} = \frac{\tau}{1+K}, \qquad t_s \approx 4\tau_{cl} = \frac{4\tau}{1+K}$$

As $K$ increases, $\tau_{cl}$ decreases, so the response reaches its final value faster. Example ($\tau = 1$ s): $K = 1$ gives $\tau_{cl} = 0.5$ s; $K = 9$ gives $0.1$ s.

Second-order system

For $G(s) = \dfrac{K}{s(s+a)}$: $\omega_n = \sqrt{K}$. Rise time and peak time $t_p = \pi/(\omega_n\sqrt{1-\zeta^2})$ decrease as $K$ rises (though overshoot increases).

Hence speed of response increases with gain.