FlightInsight
Filing Flight Plan
At the airport elevation it will be correct. If you dug a big hole to get down to sea level, the reading would be off there. The barometer setting given by the airport is the "correction" to get the altimeter to read airport elevation at the airport.
This is an interesting idea. If you dug a hole at the airport down to sea level and put a barometer at the bottom, what would it read? For us sea level pilots, this may be tough to answer, but let's imagine we're in Denver (the mile high city).
If the altimeter setting is the same as the atmospheric pressure at the airport, we'd expect altimeter setting at Denver to be about 5 inches less than a sea level setting. Check your METARs, but here is what mine say at current writing:
KCGS (College Park MD ~ sea level) - 30.05
KDEN (Denver, CO - over 5000 feet) - 29.67
Denver's a little lower on the pressure, but not by five inches. Why? Because the altimeter setting is not barometric pressure. It's barometric pressure, corrected as if it were at sea level. So if you dug a hole in Denver and put your barometer down to sea level, it should read around 29.67.
So, at sea level, temp doesn't affect altitude. Because, as others have said, the reported altimeter setting takes all this into account. But as you climb, the altimeter setting account for temperature effects at higher altitudes since we're assuming sea level pressures.
If I was in Colorado I'd do a quick vid showing errors at low temps, but the biggest mountain in my neighborhood, Sugarloaf, is barely 1,000 feet