<p>In the thin wall approximation, we study a class of asymptotically AdS black holes which contain a spherically symmetric vacuum bubble with a different (positive or negative) cosmological constant. Collapsing, expanding, and static bubble solutions are considered. Among these, expanding bubbles with positive cosmological constant can provide a way to apply the AdS/CFT correspondence to describe the physics of an expanding universe. We systematically study the causal structure of the solutions as a function of the cosmological constant, the mass of the black hole, and the tension of the bubble. We then compute the holographic entanglement entropy for a class of boundary subregions using extremal codimension-two surfaces as a probe. For collapsing bubbles, we find examples in which the entanglement entropy also explores the geometry inside the black hole bifurcation surface. As a complementary way to probe the interior of the bubble, we investigate almost-null radial geodesics related to the bulk-cone singularities of boundary two-point correlators. While the bulk-cone singularities for collapsing and expanding bubbles are consistent with thermalization at late time, static bubbles violate thermalization and exhibit properties similar to those of scar states.</p>