What I expected to happen was...:
You should mine a Stone Slab with an axe just as fast/slow as with nothing selected/barehanded and the slab breaks.
What actually happened was...:
Axes mine Stone Slabs faster, but breake them. Efficiency also applies, and the tool quality (Wood, Stone, Iron, Diamond) matters.
Steps to Reproduce:
1. Put down a Stone Slab
2. Equip an axe
3. Watch as it goes faster than with your hands, but it breaks.