Because what you are doing is a flow process, with mass inflow and no mass outflow, you need to use the thermodynamic equation:
$dU_{cv}={m_{in}d}{H}_{in}-{m_{out}d}H_{out}+\delta Q-\delta W_{shaft}$
If you insulate your air cylinder well enough, $\delta Q = 0$.
Assuming that your air cylinder does not deform, $\delta W = 0$.
Since you are filling your cylinder with air and assuming no air escapes, ${m_{out}d}H_{out} = 0$
Therefore, the enthalpy of the gas which you are filling adds to the internal energy of the gas in the cylinder, and because the internal energy is positively correlated to temperature, the temperature in of the gas in the cylinder rises.
$\Delta{U_{cv}}={H_{in}}>0$, so
$\Delta{T} >0$
You may apply the reverse for the release of air from the cylinder. In this case:
$\Delta{U_{cv}}={-H_{out}}<0$, so
$\Delta{T} <0$
http://en.wikipedia.org/wiki/Thermodynamic_system#Flow_process
When you compress a gas into a deodorant can it does indeed heat up. Then if you let the can cool the pressure falls again. The pressure and temperature will remain related by (approximately) the ideal gas law:
$$ P = \frac{nR}{V}\,T $$
You say in your question:
It seems like the gas equation doesn't apply when the gas is cooling down within the can and the pressure remains the same. (my emphasis)
But the pressure doesn't remain the same while the can is cooling.
For completeness we should note that in real deodorant cans the gas used liquifies under pressure so the behaviour is more complicated than a simple ideal gas.
Best Answer
Well the real question should be why is there a °C (Celsius).
The Celsius scale is a "Centigrade" scale in that it uniformly divides the temperature range between the boiling point of water, and the freezing point of water into 100 equal parts, and then it arbitrarily calls the freezing point zero °C, and the boiling point becomes 100°C.
The Kelvin scale is referenced to the triple point of water, not the freezing point, and that Temperature is about 0.1°C (it might be 0.098°C but I am not sure about that).
Quite arbitrarily, it was decided that degrees on the Kelvin scale, should be identical in size to Celsius degrees, and experimentally the zero on the Kelvin scale (zero kelvins) is 273.16 Celsius degrees below the triple point of water, which makes it also -273.15°C