Which year is your Mustang, and how many miles does it have? Where is the coolant temperature gauge needle? All the way to the H mark, or just beyond the middle?
So your Mustang was working fine before the gear swap? The only thing related to a gear swap I can think of is the higher engine load under acceleration, and the higher RPM, after gear swap. They shouldn't touch your engine for this work.
Is your coolant level good? Did you check to see if there is any engine code? I wonder if your thermostat is stuck closed.
2011. 34,000 miles. just went through dealer inspection no issues. all the way to H.
Modifications Performance: Airaid CAI, 93P Tune by Torq, Dynomax Exhaust Interior: Auto Headlamp Switch, Leather Seat Upgrade, Infinity Speakers, Billet Chrome: Door Locks, Radio Display Trim, Door Lock/Unlock Covers, Map Light Button Covers, Air Vent Covers Exterior: Chrome Door Handles, Faux Gas Lid Delete; 3.7 Emblems, Stock Wheels Painted Gloss Black, Tinted Lights, Dark Gray Rocker and Hood Crease Stripes, Upper and Lower Billet Grille Overlays Engine Compartment: Custom Strut Brace