The equations you need here are CAS is proportional to ½ rho(density) v2 (v is TAS)

And

Mach No =TAS/LSS (local speed of sound)

Stated in question that Auto throttle is to maintain a constant CAS.

If outside air temperature increases, the density decreases. To keep CAS constant the value of V (TAS) must therefore increase.

Using the second equation, as temperature increases the LSS also increases. With both LSS and TAS increasing the mach number remains constant.