Gnuplot: Plot x2 axis with respect to x1 axis

The best way: Plot your data as usual on the x1 and y1 axes, but place additional labels on the x2-axis with x2tic(column_number):

set x2tics
set xtics nomirror
plot 'data.txt' using 3:2:x2tic(1) w lp

see: Plot y1 in x1 with respect to x2 axis

Here is how you can achieve this. I first show you the script and the result, and later explain the steps:

reset
set xtics nomirror
set x2tics
set autoscale xfix
set autoscale x2fix
set xlabel 'P'
set ylabel 'T'
set x2label 'V'
plot 'data.txt' using 3:2 with linespoints ps 2 lw 2 title 'T wrt P', \
     '' using 3:2:x2tic(1) axes x2y1 with points ps 2 lw 2 title 'T wrt V'

I first plot T wrt P on x1y1. Afterwards I plot T wrt V on x2y1 and use for this the range and tic positions of P, but use the V values as tic labels for the x2 axis. This gives a linear scale for P and adapts V accordingly.

In order for this to work you must use set autoscale xfix and set autoscale x2fix. This uses the exact ranges and does not expand an axis to the next major tics, which would be done only for the x axis, but not for the x2 axis, which has custom tics.

You could of course also reverse the process and use a linear scale for V and adapt the P tics. In any case, for the custom tics, which are placed with xtic() or x2tic, the numbers are used like they are formatted in the data file.

reset
set xtics nomirror
set x2tics 1
set autoscale xfix
set autoscale x2fix
set xlabel 'P'
set ylabel 'T'
set x2label 'V'
plot 'data.txt' using 1:2:xtic(3) with linespoints ps 2 lw 2 title 'T wrt P', \
     '' using 1:2 axes x2y1 with points ps 2 lw 2 title 'T wrt V'

Here, the points are shown for both plot lines, to demonstrate, that they really coincide.

In order to have the one command only generating the xtics, one can use NaN for the y-value. And if only some of the custom tics should be labels, one needs an appropriate check in the x2tic call. Here, I set labels only for all even rows $0 is the current row number, starting from 0):

reset
set xtics nomirror
set x2tics
set autoscale xfix
set autoscale x2fix
set xlabel 'P'
set ylabel 'T'
set x2label 'V'
plot 'data.txt' using 3:2 with linespoints ps 2 lw 2 title 'T wrt P', \
     '' using 3:(NaN):x2tic((int($0) % 2) ? '' : stringcolumn(1)) axes x2y1 t ''

With the result: