Silicon quantum dot qubits show great promise but suffer from charge noise with a 1/fα spectrum, where f is frequency and α≲1. It has recently been proposed that 1/fα noise spectra can emerge from a few thermally activated two-level fluctuators in the presence of sub-bath temperature fluctuations associated with a two-dimensional electron gas (2DEG). We investigate this proposal by performing Monte Carlo simulations of a single Ising spin in a bath with a fluctuating temperature. We find that to obtain noise with a 1/fα spectrum with α≲1 down to low frequencies, the duration of temperature fluctuations must be comparable to the inverse of the lowest frequency at which the noise is measured. This result is consistent with an analytic calculation in which the fluctuator is a two-state system with dynamics governed by time-dependent switching rates. In this case we find that the noise spectrum follows a Lorentzian at frequencies lower than the inverse of the average duration of the lowest switching rate. We then estimate relaxation times of thermal fluctuations by considering thermal diffusion in an electron gas in a confined geometry. We conclude that temperature fluctuations in a 2DEG sub-bath would require unphysically long durations to be consistent with experimental measurements of 1/f-like charge noise in quantum dots at frequencies extending well below 1 Hz
