  This is the first in a series of two papers studying mu-cscK metrics and
muK-stability, from a new perspective evoked from observations in
arXiv:2004.06393 and in this first article.
  The first paper is about a characterization of mu-cscK metrics in terms of
Perelman's W-entropy $\check{W}^\lambda$. We regard Perelman's W-entropy as a
functional on the tangent bundle $T \mathcal{H} (X, L)$ of the space
$\mathcal{H} (X, L)$ of K"ahler metrics in a given K"ahler class $L$. The
critical points of $\check{W}^\lambda$ turn out to be $\mu^\lambda$-cscK
metrics. When $\lambda \le 0$, the supremum along the fibres gives a smooth
functional on $\mathcal{H} (X, L)$, which we call mu-entropy. Then
$\mu^\lambda$-cscK metrics are also characterized as critical points of this
functional, similarly as extremal metric is characterized as the critical
points of Calabi functional.
  We also prove the W-entropy is monotonic along geodesics, following
Berman--Berndtsson's subharmonicity argument. Studying the limit of the
W-entropy, we obtain a lower bound of the mu-entropy. This bound is not just
analogous, but indeed related to Donaldson's lower bound on Calabi functional
by the extremal limit $\lambda \to -\infty$.
