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This is the second in a series of two papers studying $\mu$-cscK metrics and$\mu$K-stability from a new perspective, inspired by observations on$\mu$-character in arXiv:2004.06393 and on Perelman's $W$-entropy in the firstpaper arXiv:2101.11197.  This second paper is devoted to studying a non-archimedean counterpart ofPerelman's $\mu$-entropy. The concept originally appeared as $\mu$-character ofpolarized family in the previous research arXiv:2004.06393, where we used it tointroduce an analogue of CM line bundle adapted to $\mu$K-stability.  We firstly show some differential of the characteristic $\mu$-entropy$\mathbf{\check{\mu}}^\lambda$ is the minus of $\mu^\lambda$-Futaki invariant,which connects $\mu^\lambda$K-semistability to the maximization ofcharacteristic $\mu^\lambda$-entropy. It in particular provides us a criterionfor $\mu^\lambda$K-semistability working without detecting the vector $\xi$involved in the $\mu^\lambda_\xi$-Futaki invariant.  In the latter part, we propose a non-archimedean pluripotential approach tothe maximization problem. In order to adjust the characteristic $\mu$-entropy$\mathbf{\check{\mu}}^\lambda$ to Boucksom--Jonsson's non-archimedeanframework, we introduce a natural modification$\mathbf{\check{\mu}}^\lambda_{\mathrm{NA}}$ which we call non-archimedean$\mu$-entropy. We extend the non-archimedean $\mu$-entropy from the set of testconfigurations to a space $\mathcal{E}^{\exp} (X, L)$ of non-archimedean pshmetrics on the Berkovich space $X^{\mathrm{NA}}$, which is endowed with acomplete metric structure. We introduce a measure $\int \chi\mathcal{D}_\varphi$ on Berkovich space called moment measure for this sake,which can be considered as a hybrid of Monge--Amp\`ere measure andDuistermaat--Heckman measure.

Entropies in $\mu$-framework of canonical metrics and K-stability, II --  Non-archimedean aspect: non-archimedean $\mu$-entropy and  $\mu$K-semistability