Why is the action of phagocytes considered a nonspecific response?

It does not occur at any specific temperature.

It is not involved with the production of any specific type of lymphocyte.

It does not respond to any specific type of hormone.

It is a response to any type of harmful substance.

The best answer is that it does not involve the production of any specific type of lymphocyte.

When a pathogen breaches the skin and enters the body, it meets the first line of (systemic) defense inside the body, namely white blood cells called neutrophils. Neutrophils are always in circulation in the blood and when they meet any foreign body, they attack it by engulfing it (phagocytosis) and kill it.

Neutrophils do not have memory of any pathogen. They attack any pathogen indiscriminately, not selectively. This is why they belong to the innate (non-specific) arm of the immune system.

Memory T - cells are lymphocytes that are specifically produced and act against pathogens that were encountered at least at one time before. The memory of that encounter is kept and when the same kind of pathogen is encountered a second time, it is remembered by these cells and they very specifically single it out and destroy it. These cells belong to the specific or adaptive arm of the immune system.