Shouldn't enthalpy be constant measured at constant volume?
Enthalpy is heat flow associated with a constant pressure process.  Here is how:

Start with the familiar first law of thermodynamics for energy change:

$\Delta E=q+w$

$E$=energy

$q$=heat flow into the system

$w$=work done on the system

The work done is rendered thusly:

$w=-\int_\text{start}^\text{end}P\,\mathrm dV$, $P$=pressure, $V$=volume

Pressure is deemed constant so:

$w=-\int_\text{start}^\text{end}P\,\mathrm dV=-P\Delta V=-\Delta(PV)$

And then our first law balance equation becomes:

$\Delta E=q-\Delta(PV)$

$q=\Delta(E+PV)$

So heat flow is the change in a combination of state variables involving energy, pressure and volume.  This combination is what we define as enthalpy and usually label $H$.  Thereby, this enthalpy variable is designed to automatically include (mechanical) work inherent to a constant pressure process.