The axe is a common heraldic charge, just as the cross is a common Christian symbol. Whereas a hammer is a tool used in construction, an axe is used for destruction. And yet that destruction could be a preliminary stage in the construction of something.
The axe is one of the oldest tools developed by humans; since the Neolithic age, it has been a symbol of battle and work. It can be a tool and a weapon of death.
Christianity associates the axe with few martyrs (usually St. Matthew, but many others were also executed with an axe).