[Inquiry] Exhaust Temperature of Freezers/Coolers

When it comes to the coolers/freezer setup, how much does the exhaust temperature of the cooler matter? I know the outer area's temperature affects the insulation, but does a higher outer temperature actually affect the efficiency of the cooler? I'm using a freezer built in to a mountain (so overhead rock to help stabilize cooling) and a cavity wall for extreme insulation. If I pump heat into a small area and it heats up quite a lot, do my coolers fail in keeping the other extremely well insulated parts cold?