  In this article, we consider the following family of random trigonometric
polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a
given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which
are centered and standardized. We set $\mathcal{N}([0,\pi],Y)$ the number of
real roots over $[0,\pi]$ and $\mathcal{N}([0,\pi],G)$ the corresponding
quantity when the coefficients follow a standard Gaussian distribution. We
prove under a Doeblin's condition on the distribution of the coefficients that
$$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],Y)\right)}{n}
=\lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],G)\right)}{n}
+\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)-3\right). $$ The latter establishes
that the behavior of the variance is not universal and depends on the
distribution of the underlying coefficients through their kurtosis. Actually, a
more general result is proven in this article, which does not require that the
coefficients are identically distributed. The proof mixes a recent result
regarding Edgeworth's expansions for distribution norms established in
arXiv:1606.01629 with the celebrated Kac-Rice formula.
