.... and if so how hard can it be hardened? How hard is it in its natural condition?
Don't think its hardenable by the normal methods as used for steel or heat treatable aluminums.
I've got some aircraft quality rotary forged rings made of 6-4 that check Rc33 fairly consistently. 6-4 is one of the more common alloys. CP titanium is most likely softer than this.
Titanium is easily work hardened........sometimes too easily. Try drilling a hole in it with a slightly dull drill and then tap it. The tap will probably stick and or break. (small taps anyway.)There are some alloys that will age harden. There are Ti alloys that can be heat treated to modify their tensile and ductile properties.
ASTM has a lot of data on the various alloys of Titanium.
Many titanium alloys are age-hardened along the same lines as, say 6061 aluminum. You heat it above a certain temperature to put alloying elements into solution in the metal. (It's not molten; it's called a solid solution.) This is called solution treating. Then you quench to freeze the solution in place. Unlike carbon steel, the quenched titanium alloy is now soft. Next, you re-heat at a lower temperature for a controlled time to let some of the dissolved constituents precipitate out. This hardens the metal, and is called age hardening. The strength increases with time and temperature, peaks, then begins to fall off, until you finally reach the annealed state, and the alloy is soft again. The reason it is called age-hardening is that some alloys, like 2024 aluminum, will actually harden with age at room temperature. It just takes a long time. Most of the time, you will want to help the ageing process with heat. Books like _ASM Metals Handbook_ tell you times, temepratures, etc., for a given alloy. It won't work with all alloys or commercially pure titatnium. Beware that heat treating titanium ought to be done in a controlled-atmosphere or salt pot furnace to avoid certain issues like embrittlement from contact with air at temperature.