It is a partial answer for your question:

the normalized laplacian spectrum of $P_n$ for $n=2,3,4,5$ are:

$P_2:$ $[0,2]$,

$P_3: [0,1,2]$,

$P_4: [0,2,\frac{1}{2},\frac{3}{2}]$,

$P_5: [0,1,2,1-\frac{\sqrt2}{2},1+\frac{\sqrt2}{2}]$.

as you can see, the normalized laplacian eigenvalues are simple.

In general, for $P_n$, the form of normalized laplacian matrix is very special and its eigenvalues are simple.