1) What kind of thermometer do you have in mind which would be able to resolve temperature variations on the picosecond time scale?

2) If in a solid typical phonon frequencies are of order, say \hbar*omega ~ 1..10meV, then the corresponding classical vibrational frequencies amount to 0.4..4ps (hope I did the math right). So what kind of a process do you imagine which can tell you about temperature on this very same timescale? When does it even make sense to talk about temperature?

3) I think to discuss that properly you must include a description of both the object whose temperature is to be determined and the thermometer (including the measurement principle, as such [by what mechanism is the temperature of the thermometer being identified] and in conjunction with the object).

Finally, what is your notion of temperature fluctuation at all? What is the context? A canonical state of fixed temperature is one in which energy fluctuates. On a formal level temperature may be defined as the derivative of entropy with respect to energy. In what sense do you want us to understand the notion of a fluctuating temperature?

To give a second answer: if the thermometer is in equilibrium with a heat bath of some temperature, then by definition the temperature is constant and not fluctuating.

Whatever is your READING of the temperature (e.g. the length of the piece of glass) may fluctuate, though. If it is macroscopic, then expect any fluctuation to scale, in one form or another, as the square root of the number of unit cells. By all normal means, that will be pretty small for a macroscopic thing. How small at the ps time scale I don't know.

In order to measure temperature, the thermometer must be in thermal equilibrium with the heat bath. For electrical insulators, such as glass, at low temperatures phonons are the agents solely responsible for bringing about thermal equilibrium (must add that in metals, electrons come in thermal equilibrium long before ions reach this state). For this to take place, sound waves must have had time to have travelled through the system (here a piece of glass) a number of times. Taking an idealised model of a thin piece of flat glass of thickness d, the equilibration time is equal α d/v_s, where α is a constant of order 10, and v_s the sound velocity. Sound velocity in glass at around room temperature is equal to 3962 m/s. Thus at around room temperature sound travels through the thickness of a piece of glass with d = 1 mm in some 0.25 μs. Identifying α with 10, the minimum time required for measuring temperature with a thermometer whose glass thickness is 1 mm amounts to some 2.5 μs.

Above I have referred to sound velocity at around room temperature. The temperature dependence of sound velocity is an important fact to take into account. As well-known, in an ideal gas sound velocity goes to zero like θ^{1/2}, where θ is the absolute temperature. In general, the lower the temperature, the longer it takes for the equilibrium to set in in the glass I discussed above.

The glass absorber will have a certain thermal time constant and probably getting down to the picosecond timescale would be difficult.

The best way to measure temperature at fine scales is to exploit Brownian motion. Whether that is possible or not will depend on your sample (which you haven't specified in the question).

But if you can get an ensemble of Brownian particles to be in equilibrium with your sample, you can video them. Then from the video the velocities and sizes of the particles can be estimated and you can statistically estimate temperature.

What's neat about this method is that you can also estimate the uncertainty in the temperature. I suspect you'll find that the shorter the timescale you measure the higher the uncertainty in temperature.

Because your glass thermometer is a macroscopic body, then according to the ergodic principle the temperature readings over a short time would not result in non-fluctuating results, right? Even if the time interval is as short as pico seconds?