Power MOSFET gets hot while driving Peltier by step-down buck voltage converter
To protect against possible future edits, here is the circuit we're talking about:



Major problem are:

There are no component designators in the schematic, making the circuit hard to talk about.

The 1 mF cap makes absolutely no sense.  You have a double emitter follower driving the FET gate.  This provides substantially more current capability than the digital output.  OK so far.  However, the cap prevents sustained DC into the power input of this current amplifier.  I can't even guess what you think it's purpose is, but it keeps the current amplifier from working properly.

There is also no need for the 3.3 &Omega; resistor.  Just connect the collector of the NPN directly to the 12 V supply.

The FET isn't getting nearly enough gate drive voltage.  The double emitter follower current amplifier is a reasonable idea in itself, but it looses two junction drops of voltage.  You start with a 0-5 V digital signal, and the FET is driven with about 700 mV to 4.3 V.

That would be OK if the FET was solidly on with 4.3 V on the gate, but it's not.  You have to actually read the datasheet.  On page 2 it shows the on resistance as being 17.5 m&Omega;, but at 10 V gate drive.  On the very next line down, it shows the gate threshold can be as high as 4.0 V, at which point you only get 250 &micro;A.

The diodes is labeled "Schottky", but the symbol says ordinarly silicon diode.  Which is it?  It should be a Schottky.