Gases and physical chemistry
I will try to summarize what Karl is saying about the ideal gases.
Suppose that you measure the temperature of a gas contained in a canister. Suppose that you measure the temperature $T_i$

Using your equation that you derived from $PV=nRT$, we should have $$T_i=\frac{P_iV_i}{nR}$$
Suppose that we then apply work to the canister. Since it is an ideal gas the only way that we can change its energy is through a change in volume ,pressure, or number of moles. So that is what we are going to do. Suppose that we compress the canister so that the pressure of the gas changes. Thus, our new canister will have a new pressure $P_f$ and a new volume $V_f$.
This volume $V_f$, however, is related to $V_i$ through the ideal gas equation $$P_iV_i=P_fV_f$$
Hence we have that $$V_f=\frac{P_iV_i}{P_f}$$ 
Lets substitute this equation to find the new temperature of our compressed canister:

$$T_{compressed}=\frac{P_f\frac{P_iV_i}{P_f}}{nR}$$
Which gives us the initial equation. Hence the temperature of the ideal doesn't change when it is compressed. 

I will post on what happens when you increase the number of moles.