There are a couple different aspects here (well, probably more than just a couple…).
Fourier analysis is a pair of operations, where you transform bases from time to frequency and vice versa.
- the analysis equation describes how you take an entire time series, and integrate/sum over it to get the estimate of one frequency coefficient magnitude and phase. “Analysis” comes from the Greek for cutting—we often picture this as cutting up a time series into its component frequency parts.
- the synthesis equation describes taking one entire frequency spectrum (with both frequency and phase information per frequency) and estimating one time point value. “Synthesis” comes from the Greek for combining or putting together—we often picture this as summing up the components to better approximate and reconstruct a time series.
(Sidenote that the Fourier relations are essentially symmetric, so we could have started from a frequency point of view for naming, but we don’t.)
But the above means that, to estimate the power spectrum of a single frequency, you need to integrate or sum over an entire time series. So, if you chop away time points, by necessity you are changing your estimate of a frequency.
That being said, if your entire time series has essentially constant frequency/statistical content throughout, for example being pure white noise or repetitive over very small time scales throughout, then chopping away initial time points might not actually affect your final coefficient magnitude values very much over a given interval of the frequency spectrum on average.
But with that being said, note that the exact physical frequencies you estimate values for depend on the number of time points you have (and the sampling rate, though let’s assume that is constant). The maximum frequency you can estimate with Fourier relations (for the unique band that includes the baseline) is determined by the sampling rate (so, here, constant), and it is called the Nyquist frequency, Fnyquist. But the number of frequencies between 0 and Fnyquist that you can estimate is determined by your number of time points. So, if you change the length of your time series slightly, you are estimating slightly different frequency components. (These are called harmonics, because they are factors of a fundamental frequency.)
And also, power spectrum estimates of our noisy datasets should also be considered statistical estimates—it’s not just the estimate that should matter, but the uncertainty associated with it, as well. In many applications, people don’t just estimate power of a single frequency, with might be a bit unstable (i.e., have high uncertainty), but instead of some range of frequencies, to have smaller uncertainty/more stability. From that point of view, if you are averaging over frequency windows/bands, and your time series has constant statistical properties throughout, then chopping away time points might not matter so much.
This has been a long answer, but it is not a question that has a simple yes/no. Constancy of frequency properties depends on what exactly you are measuring, choosing a stable way to measure it (i.e., using a frequency window, not single frequency), and the details of the time series themselves.