In reality, there are only a couple of things that happen to espresso over the course of a few minutes. First, the crema or foam that sits on top of the espresso reintegrates with the liquid. Crema, on its own, contains some of the best flavors and aromatic qualities of the espresso in high concentrations. Drinking it straight can be overwhelming but, it stands to reason, that having the crema reintegrate might actually be a good thing. Second, the espresso cools down. The temperature of espresso has a lot to do with how we taste the various flavors contained within it and, in fact, our taste buds have many microscopic channels called TRPM5 that increase the bud's sensitivity at higher temperatures. If, however, espresso passes a certain temperature (the exact temperature being unique to every person) it will burn our tongues and greatly decrease our ability to taste. Given that espresso typically comes out of the machine at about 200 degrees Fahrenheit, a fresh shot can easily burn off our ability to taste its liquid-y goodness or numb us to its heave-worthy horrible-ness. In essence, a shot that has sat for about a minute may actually be a better representation of the espresso than a fresh one.
I don't know how the rumor of "dead shots" got started but I do know if you work at a shop that serves poor coffee (or poorly serves great coffee) it would be advantageous to let people drink the shots hot. That way their burnt taste buds won't be able to tell them that they're drinking something awful.
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