My dog ran away because the neighbor left the gate opened.

Most states limit recovery to the value of the dog and grant nothing for sentimental value.

While you have to right to sue, it is unlikely worth the effort. Dogs are considered personal property and you are not likely to be awarded anything above the sales value of the dog. Unless this is an unusually expensive dog, the lawsuit will probably cost more than your recovery.

I agree you will get the retail value of the dog which is likely not even the cost of the filing fee; courts do not value emotional loss of a pet. It would not be in your best interest to sue here.