Transmit power of non-orthogonal FDM
In your example, the sinusoids are still orthogonal to one another; the key is that you must integrate over the correct period when calculating $P$. All of the sinusoids have frequencies that are multiples of $\frac{\phi}{T}$, so their resulting sum will have period $\frac{T}{\phi}$.

You should therefore take this fact into account when calculating the average power over a period of $s(t)$:

$$
P = \frac{\phi}{T} \sum_{k = 0}^{N-1} \sum_{k' = 0}^{N-1} a_k a^*_{k'} \int_{0}^{\frac{T}{\phi}} e^{j2\pi \phi (k - k')t/T} dt
$$

With this formulation, then you do get the useful property that the integral is zero for all $k \ne k'$, which allows you to simplify the expression to be simply the sum of the powers of each of the sinusoids.