Specific Heat affected by Pressure I know the specific heat depends (slightly) on the conditions present. For example, methane's specific heat is 2.5 at 400 K, but 4.5 at higher temperatures. 
How might specific heat depend on pressure, though. -- if we are under super low pressure (say 6 mbar rather than earth's typical 1014 mbar), how would specific heat change. I am expecting it to differ along with $PV = nRT$, but not sure. 

Also, $\Delta H = mc\Delta T$ -- would I need to plug $\Delta PV/nR$ in for $\Delta T$?
At low pressures like 6 mb and even 1014 mg, the heat capacity is very insensitive to pressure (and the ideal gas value can be used).  At higher pressures, where non-ideal gas effects set in, the enthalpy and the heat capacity are, in general, both functions of pressure as well as temperature.  The partial derivative of enthalpy with respect to pressure at constant temperature for an arbitrary substance is given by:

$$\left(\frac{\partial H}{\partial P}\right)_T=\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]$$
This is equal to zero for an ideal gas.