I mean that'd be like nearly 400 MG of caffeine. That's the max amount that the FDA considers safe to consume in a day. Drinking all that in the morning would be excessive and likely to make you jittery and anxious. I'd recommend against doing that.
Not a great idea, but not for the caffeine quantity that others mention (though perhaps for that, too). Espresso is meant to be drunk quickly and hot. Even the glasses are designed to keep the surface area minimal and give the drinker control over how (minimally) cool they allow the espresso to get.
Keeping a large amount in a mug for a while will allow bitter notes to develop and will mute the creamy and sweet notes/feel that make espresso worthwhile. You can minimize that a bit by diluting the coffee (a 3-hour old latte or americano changes less than a straight shot would in the same span), but it'll still change.
Pour over is really the only method that is intended to be drunk at varying levels of heat, and you can make it super strong if that's what you're looking for.
I mean, it's just a lot of caffeine. I'm sure some people have no problem, but others will be off the walls or just sick. What's your typical caffeine intake? If you just want a hot drink to sip on for longer, try a couple shots of espresso and top the mug off with hot water for an Americano.
354ml of coffee is quite a bit, even with strong robusta local coffee I don't do more than 180ml on average, and 4-5 shots sounds like a lot. I suppose if your body can take it, that's fine, but I do wonder if it'll taste like ass after an hour or so; won't it get cold?
My regular morning brew is a 300ml moka pot in a mug with frothy milk. According to this I'm taking about 516mg of caffeine each morning, without spacing it out.
I have owned that thing for over a decade and have always treated it as a single serving device. I think FDA's wisdom is mostly a guideline, and nothing beats simply knowing your own tolerance.