great idea !

concerning the deviations from the ideal gas law i have some suggestions why it does not work:

1: sensor inertia especially of the thermometer as already suggested by fbouquet:

you can see it in the time series of T(P) : it falls and raises faster than T(measured) because the thermometer needs time to adapt to the envirnoment.

There is a way to correct that (see below)

2: Temperature sensitivity of the pressure sensor: pressure sensors are rather temperature sensitive and usually need a temperature correction. Could be that your sensor tag does a T-correction, but you never know. A check of the Temperature senstivy would be interesting: put the thing into the fridge with the jar open, and the door of the fridge not fully closed (i know this is not energy efficient - but we have to ensure that pressure remains constant in the fridge during the experiment)

3: Is your jar 100% airtight ?

... i think it is airtight ohterwise the T(P) curve would underestimate temperature (... if i am right)

But in principle this is a good exercise to test the understanding of an experimental setup.

Inertia correction:

Lets say a thermometer gives a Temperature Ts. If it has some inertia such that it adapts to a sudden change in temperature with a tiime constant tau i.e. after a time tau it as covered 1/e of the temperature difference. This behaviour of the thermometer can be described by the differential equation

dTs/dt = -1/tau (Ts-Te) (eq1)

where Te is the environmental temperature you want to know.

On may discretize the differential equation (There is also an analytic, more accurate method to solve the equation but for this forum and the target of phyphox this might be sufficient.)

(Ts(t2)-Ts(t1)) / dt = -1/tau ( Ts(t1) - Te(t1) ) (eq2)

and solve for Te :

Te(t1) = (Ts(t2)-Ts(t1))*tau/dt + Ts(t1)

= tau/dt * Ts(t2) + (1- tau/dt) * Ts(t1) (eq3)

where t1 and t2 are times separated by one time step of length dt.

To use this your time step dt must be sufficiently larger than tau; you have to oversample - measure at a higher temporal resolution than your sensor can resolve.

Here is a paper that uses this (or a similar) method for humidity senors on weather ballons (=radiosondes)

https://doi.org/10.1175/1520-0426(2001)0...H>2.0.CO;2
The above equation eq3 for Te can be seen as a recursive digital filter where you use preceeding values and some weights to calculate an underlying quantitiy.

I.e. something like

y(t2) = w0 * y(t1) + w1 * x(t1) + ...

With such filters you can implement low-pass, high-pass, band-pass and many other filters.

What the sensor in terms of filtering does is damping down higher frequencies, i.e. it is a low pass filter. Equation eq3 does than the inverse : it amplifies the damped frequencies. Unfortunetaly it does this for all frequencies, also for those which contain only noise. To remove this noise it might be necessary to smooth Te(t) from eq3 with a cut off frequency fsmooth=1/tsmooth with tsmooth < tau. One can do this with any running averaging or again with a digital filter. Take eq2 which describes how the sensor smoothes the information from Te; replace Ts by Te_sm the smoothed environmental temperature and Te by Te_raw which is the output from eq3. You want to smooth Te_raw with time constant t_smooth and eq2 thus becomes

(Te_sm(t2)-Te_sm(t1)) / dt = -1/t_smooth ( Te_sm(t1) - Te_raw(t1) ) (eq4)

solve for Te_sm(t2):

Te_sm(t2) = (1-dt/t_smooth) * Te_sm(t1) + dt/t_smooth * Te_raw(t1) (eq5)

In principle it is easy to implent this in python or whatever you use for anlysis.

I tried to build a digital filter of order one with the phyphox editior but ended up with compliceted combintion of exitisng elements and failed so far.

But maybe it is an idea to include such a filter in the analysis modules