Stop in for a cup of coffee

Only sort of.

It is known as a “wide spot in the line” where fluid flow slows. Fluids are not compressible and the pressure won’t drop or increase in a closed system...only flow rate is affected.
Atf is paraffin based,it expands and contracts considerably. The volume in the cooler will be significantly greater so any temperature change wont affect flow.
The hayden coolers are designed to slow flow when fluid is cool,as it warms up,the flow increases.