
APK costs just 20 euro at any Mazda dealer since there is no emissions test to be done!
A "stock-like" battery mostly costs between 100 and 150 euro, however the dimensions are important as stated before because of the cooling (you can see the cooling airduct in the left lower corner of the battery box, if the cooling is blocked life and performance of the battery will be potentially lower).
The battery swap is very simple though and can be performed yourself.
You might want to spend some more money and upgrade your battery while at it, make sure you buy one with the right dimensions and high CCA (Cold Cranking Amps).
Good luck!