The equations you need here are CAS is proportional to ˝ rho(density) v2
And
Mach No =TAS/LSS (local speed of sound)
Stated in question that Auto throttle is to maintain a constant Mach Number as total air temperature increases.
LSS changes with temperature – an increase in temp causes an increase in LSS.
So using the second formula, if Mach number remains constant (as questions states) when LSS increases so does TAS.
Using the CAS formula. An increase in temperature will decrease rho but as we’ve agreed that TAS must increase to maintain the Mach number this means the value of CAS will remain constant.