The cosmological constant $\Lambda$ can be written as part of the T-Tensor. It can then be considered as vacuum energy ($\rho_{vac}$) and vacuum pressure($p_{vac}$). $\rho_{vac}$ and $p_{vac}$ are the entries of the T-tensor in its diagonal, $T_{00}=\rho_{vac}$, $T_{11}=T_{22}=T_{33}=p_{vac}$. Because of the different sign of the time dimension it leads to $\rho_{vac}= -p_{vac}$.
Using $\Lambda$ in the field equations, vacuum (the space itself) gets a positive energy and a negative pressure.
For particles, $\rho_{vac}$ and $p_{vac}$ are independent from each other. Aren't they?
Why does no one complain when the vacuum energy density $T_{00}$ and the vacuum pressure are (with the introduction of $\Lambda$) restricted to have the same value?
Couldn't the energy density of the vacuum be totally independent from the pressure of the vacuum? Aren't those two totally different characteristics: The one is just the energy of the vacuum, the energy of space itself, the other is how space expands? Shouldn't there be two cosmological constants? One for $p_{vac}$ and one for $\rho_{vac}$? Are there any papers using two different cosmological constants?
Or is it clear from other sources than a definition that $p_{vac} = -\rho_{vac}$?