  We consider generalized Paley graphs $\Gamma(k,q)$, generalized Paley sum
graphs $\Gamma^+(k,q)$, and their corresponding complements $\bar \Gamma(k,q)$
and $\bar \Gamma^+(k,q)$, for $k=3,4$. Denote by $\Gamma = \Gamma^*(k,q)$
either $\Gamma(k,q)$ or $\Gamma^+(k,q)$. We compute the spectra of
$\Gamma(3,q)$ and $\Gamma(4,q)$ and from them we obtain the spectra of
$\Gamma^+(3,q)$ and $\Gamma^+(4,q)$ also. Then we show that, in the
non-semiprimitive case, the spectrum of $\Gamma(3,p^{3\ell})$ and
$\Gamma(4,p^{4\ell})$ with $p$ prime can be recursively obtained, under certain
arithmetic conditions, from the spectrum of the graphs $\Gamma(3,p)$ and
$\Gamma(4,p)$ for any $\ell \in \mathbb{N}$, respectively. Using the spectra of
these graphs we give necessary and sufficient conditions on the spectrum of
$\Gamma^*(k,q)$ such that $\Gamma^*(k,q)$ and $\bar \Gamma^*(k,q)$ are
equienergetic for $k=3,4$. In a previous work we have classified all bipartite
regular graphs $\Gamma_{bip}$ and all strongly regular graphs $\Gamma_{srg}$
which are complementary equienergetic, i.e.\@ $\{\Gamma_{bip},
\bar{\Gamma}_{bip}\}$ and $\{\Gamma_{srg}, \bar{\Gamma}_{srg}\}$ are
equienergetic pairs of graphs. Here we construct infinite pairs of
equienergetic non-isospectral regular graphs $\{\Gamma, \bar \Gamma\}$ which
are neither bipartite nor strongly regular.
