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We consider random trigonometric polynomials of the form \[f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where$(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussianprocesses with the same correlation function $\rho: k \mapsto \cos(k\alpha)$,with $\alpha\geq 0$. We show that the asymptotics of the expected number ofreal zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in thecase of independent or weakly dependent coefficients. More precisely, for all$\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $\alpha \geq 0$and $n\geq 1$ large enough such that$$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}-\ell\right|\leq\varepsilon,$$ where $\mathcal N(f_n,[0,2\pi])$ denotes the number of realzeros of the function $f_n$ in the interval $[0,2\pi]$. Therefore, this resultprovides the first example where the expected number of real zeros do notconverge as $n$ goes to infinity by exhibiting a whole range of possible limitsranging from $\sqrt{2}$ to 2.

New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients