I don't think the ADC is 'cooling down'. I think the two reads may be for a different purpose.
There is only one ADC on an Arduino, and part of analogRead's job is to choose which pin to sample.
The ADC has a 'sample and hold' circuit, which holds the voltage steady while the ADC is calculating its value. You can likely imagine that it might be a problem if the input voltage is wiggling around while the ADC attempts to measure it, and the sample & hold circuit avoids that problem. (The input voltage wiggling around is even worse a problem than it might appear as the ADC does multiple estimates, and by successively improving on its previous approximation, it calculate the digital value of the input voltage.)
The sample and hold circuit uses a capacitor (think rechargeable battery) which needs some time to match the input voltage, just like it takes time to recharge a battery. How long it takes depends on how much current the thing being sampled can supply. If it can't supply much current, the ADC might start measuring the sample & hold circuit before the sample and hold has matched the external voltage being measured. Reading the value twice, with a delay between, should get the sample & hold closer to the external voltage, and hence the second value should be a bit more accurate.
So, I think the two successive analoReads may to get the input voltage stable on the sample and hold.
There is a further complication. I seem to recall the very first ADC reading on an Arduino takes longer than normal, while it is stabilising, and self-calibrating. So it may be someone wants to ensure that is dealt with.
The most significant time for the Maple's ADC is not necessarily the 'conversion time', i.e. the time the ADC uses to convert an analog voltage into a number, it can do that in a microsecond if need be. The time which may dominate is the sample & hold time, i.e. the time it takes for the sample & hold circuitry voltage to come to match the external voltage. Have you any control over the external electronics? Do you know approximately what its output impedance is (its ability to supply current)?
The other point to note is, the STM32F103RB on the Maple has two Analog to Digital Converters. So it is theoretically capable of converting at upto 2,000,000 samples/second. The current Maple library only uses one of the ADC's, but there is nothing to stop you using the other.
The current Maple library allows the sample & hold circuit to charge for almost 5us.
If the input circuit can supply current fast enough, this could be reduced significantly.
Summary. 16ms is so long a time, it isn't an issue. The two analogReads are either to get a consistent read time (having got past the initial stabilisation time), or to get the sample & hold charged and stable.
