We show that the logarithmic Sobolev inequality of the Boolean cube is stable under random monotone censoring. More precisely, if $A_n\subseteq \{0,1\}^n$ is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on $A_n$ is of order $n$ with high probability. As a consequence, several analytic and probabilistic properties of the Boolean cube persist for a typical monotone subset: the censored semigroup is hypercontractive, the uniform measure on $A_n$ satisfies Gaussian concentration for Lipschitz observables, and the associated walk mixes in time $O(n\log n)$. The latter proves a conjectured mixing bound of Ding and Mossel for almost all monotone sets. The result is genuinely typical rather than universal. We construct monotone sets of density bounded away from zero whose logarithmic Sobolev constant is of order $n^2$. To prove the result, we establish a sharp logarithmic Sobolev inequality for Hamming caps and combine it with a harmonic extension argument transferring this inequality to monotone sets lying between nearby caps, together with a structural theorem of Korshunov on random monotone sets.