Let $G\subseteq U(N)$ be a Lie group. In M. Creutz, Quarks, Gluons and Lattices, 1st ed. Cambridge University Press, 2022. doi: 10.1017/9781009290395 it is claimed that the maximum of $tr\circ\cosh:\mathfrak{g}\rightarrow\mathbb{C}$ is at $0\in\mathfrak{g}$, but no proof is offered. In here the trace is taken over the defining representation on $\mathbb{C}^N$.
To prove this I tried differetiating with respect to a linear coordinate system $\omega:\mathfrak{g}\rightarrow\mathbb{R}^{\dim G}$ dual to a basis $(T_1,\dots,T_{\dim G})$ of $\mathfrak{g}$. Using the cyclic property of the trace, one has $$\frac{\partial tr(\cosh(\omega^aT_a))}{\partial\omega^b}=tr(\sinh(\omega^aT_a)T_b),$$ which clearly vanishes at $\omega=0$. However, how do I know that there are no other extrema (I don't think I really need to show that the extremum is a maximum for my purposes)?
Since you say "hyperbolic cosine" in the title I assume you meant the hyperbolic cosine in the first paragraph. In that case we can argue as follows. If $X \in \mathfrak{u}(n)$ has eigenvalues $\lambda_k$, then $\cosh X$ has eigenvalues $\cosh \lambda_k$ and hence trace
$$\text{tr}(\cosh X) = \sum_k \cosh \lambda_k.$$
Next, since $X \in \mathfrak{u}(n)$ is skew-hermitian, its eigenvalues are purely imaginary; write $\lambda_k = i r_k$ where $r_k \in \mathbb{R}$. Then $\cosh \lambda_k = \cos r_k$, which gives
$$\text{tr}(\cosh X) = \sum_k \cos r_k.$$
This function attains its maximum value of $N$ when $\cos r_k = 1$ for each $k$, which happens iff each $r_k$ is an integer multiple of $2 \pi$. This maximum is attained at the origin but it is also obtained at other places, e.g. at $X = 2 \pi i I$.