Power of an angle-modulated wave
Let's consider the complex signal
$$\phi_c(t)=Ae^{j(\omega_ct+km(t))}\tag{1}$$
The power of $(1)$ is readily computed as
$$P_{\phi_c}=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\big|\phi_c(t)\big|^2dt=A^2\tag{2}$$
IF $(1)$ were the analytic signal corresponding to the real-valued signal $\phi(t)$ then the power of $\phi(t)$ would simply be $P_{\phi_c}/2=A^2/2$.
In general $(1)$ is not analytic. However, in all practical cases of angle modulation, a sufficiently high carrier frequency makes $(1)$ analytic for all practical purposes, i.e., there are no measurable negative frequency components. Or, equivalently, a practical angle modulated signal is a bandpass signal and has no components at or close to DC. If this is the case, the power of $\phi(t)$ is indeed given by $A^2/2$.
In all other (artificial) cases, the power of $\phi(t)$ cannot generally be shown to equal $A^2/2$.