  We study the Bak-Sneppen evolution model on a regular hypercubic lattice in
high dimensions. Recent work [Phys. Rev. E 108, 044109 (2023)] has shown the
emergence of the $1/f^{\alpha}$ noise for the ``fitness'' observable with
$\alpha \approx 1.2$ in one-dimension (1D) and $\alpha \approx 2$ for the
random neighbor (mean-field) version of the model. We examine the temporal
correlation of fitness in 2, 3, and 4 dimensions. As obtained by finite-size
scaling, the spectral exponent tends to take the mean-field value at the upper
critical dimension ${\rm D}_u = 4$, which is consistent with previous studies.
Our approach provides an alternative way to understand the upper critical
dimension of the model. We also show the local activity power spectra, which
offer insight into return time statistics and the avalanche dimension.
