  Considering the negative cosmological constant of an anti-de Sitter (AdS)
background as a positive thermodynamic pressure in the extended phase space, we
study the thermodynamic behaviors of the regular Hayward-AdS (HAdS) black hole
(BH) by investigating $P-\upsilon$ critical and cooling-heating phase
transition. We compare the difference of two thermodynamic processes between
the HAdS BH and the Bardeen-AdS (BAdS) BH. It is found that the phase
transition of the BAdS BH tends to be more the van der Waals (vdW) phase
transition. For the cooling-heating phase transition, we find the inversion
curves of the HAdS BH are always higher than the BAdS BH under the same
pressure and magnetic charge. We also compare the smallest existence mass, the
zero-temperature remnant, and the critical magnetic charge. The results suggest
that the inner horizon and the outer horizon of the Hayward BH are easier to
merge, and the singularity is easier to expose.
