  Phase transition and heat engine efficiency of phantom AdS black holes are
investigated with peculiar properties found. In the non-extended phase space,
we probe the possibility of $T-S$ criticality in both the canonical ensemble
and grand-canonical ensemble. It is shown that no $T-S$ criticality exists for
the phantom AdS black hole in the canonical ensemble, which is different from
the RN-AdS black hole. Contrary to the canonical ensemble, no critical point
can be found for neither phantom AdS black holes nor RN-AdS black hole in the
grand-canonical ensemble. Moreover, we study the specific heat at constant
electric potential. When the electric potential satisfies $A_0>1$, only phantom
AdS black holes undergo phase transition in the grand-canonical ensemble. In
the extended phase space, we show that there is no $P-V$ criticality for
phantom AdS black holes, contrary to the case of the RN-AdS black hole.
Furthermore, we define a new kind of heat engine via phantom AdS black holes.
Comparing to RN-AdS black holes, phantom AdS black holes have a lower heat
engine efficiency. However, the ratio $\eta/\eta_C$ of phantom AdS black hole
is higher, thus increasing the possibility of approaching the Carnot limit.
This observation is obviously of interest. The interesting results obtained in
this paper may be attributed to the existence of phantom field whose energy
density is negative.
