The mach meter gives an indicated mach number that is the ratio of dynamic pressure to static pressure.
Mach Number = Dynamic Pressure / Static Pressure.
Dynamic Pressure = 1/2 Rho V squared
Static Pressure = Rho g H, where H is the height of the column of air above us.
So we have Mach Number = 1/2 Rho Vsquared / Rho g H
Dividing both sides of this equation by Rho gives us
Mach Number = 1/2 Vsquared / gH
Rho is not in the above equation, so changes in Rho do not change the indicated mach number.
Or to put it in a slightly different way, if the temperature changes it causes the value of Rho to change. But because Rho is in both sides of the mach number equation, this change in Rho is self-cancelling.
So changes in temperature do not affect the indicated mach number.
Also because both CAS and mach number are determined using the same dynamic pressure, the mach number at any fixed CAS does not change with temperature.
But because changes in temperature change the ratio of CAS to TAS, any change in temperature will change the TAS at any given mach number.