What does the grand sum of a linear transformation matrix tell us about its action on a symmetric positive semi-definite matrix?

Question

Let:

• $\mathbf{\Sigma}$ be an $m \times m$ symmetric positive semi-definite matrix.
• $\mathbf{X}$ be a real $n \times m$ matrix whose columns are orthonormal.
• The grand sum of $\mathbf{X}$ be $k$. I.e., $\sum_{i,j} x_{i,j} = k$.
• $\mathbf{\Phi} = \mathbf{X} \mathbf{\Sigma} \mathbf{X}^\top$.

What does $k$ tell us about the action of $\mathbf{X}$? I.e., if we know $k$ and some property or properties of $\mathbf{\Sigma}$ (value, trace, determinant, eigenvalues, etc.), what can we say about $\mathbf{\Phi}$? In particular, are there values of $k$ that have special meaning (e.g., $k = 1$)?