I'm not sure how this would look. It seems you've have to specify regime breakpoints within the data, right? Or is there some rolling method that could compare window sizes against each other?
Also key, is it possible to do this without specifying a model?
If you're fitting anything like a regression, ARMA, etc., then you might consider the LM test described in this paper (pdf link) [1]. It tests for stationarity, which is stronger than just testing for changes in variance, but perhaps you can modify for your purposes, or use that to go looking for more applicable tests.
Along the same lines, you could set up a prior distribution over the value of $\sigma^{2}_{u}$ (in the paper's notation for the random walk variance) and use your regression/ARMA/whatever to obtain a likelihood model for $P(y_{t}|\sigma^{2}_{u})$, and then use simulations to draw lots of samples from $P(\sigma^{2}_{u}|y_{t})$. Then you can use posterior predictive checking and Bayesian p-values to test whether $\sigma^{2}_{u}$ is meaningfully different than 0. If not, you have reason to suspect the variance is not changing.
I know this isn't spelled out too carefully, and there may be more "ready-to-use" statistical tests, but hopefully it gives you some ideas.
[1] Kwiatkowski et al, Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?, Journal of Econometrics, Vol. 54 Issues 1-3. 1992. DOI: 10.1016/0304-4076(92)90104-Y
