The paper is deeply flawed from both the climate science and machine learning perspectives. The most obvious being the most eye-catching claim that equilibrium climate sensitivity is approximately 0.6C, which if true would overturn our understanding of the climate system. However the paper doesn't actually explain how this figure of 0.6C is obtained from a "largest deviation" of 0.2C, it is basically just a hand-wave. Also the largest deviation is not 0.2, this is the largest average (mean absolute) deviation seen in the proxies, and you can have a mean absolute deviation without there being a trend that you could relate to increased GHG concentrations and hence estimate ECS. More importantly, this would give an estimate of transient climate sensitivity, not equilibrium climate sensitivity, and you can't reliably estimate ECS (which is global) from regional or sub-regional proxy records.
Approaches such as the one taken in this paper, which seek to see how much of the data can be attributed to "climate cycles" with the remainder being taken as the anthropogenic component are inherently biased towards low estimates of ECS. This is because of omitted variable bias; because the anthropogenic forcing signals are not included in the model, if the net effect of independent changes in the forcings is correlated with a sinusoidal component, they will be wrongly attributed to these climate cycles, when in fact they are produced by the forcings. Models like this can only be used to estimate lower bounds on ECS, likewise if you make a model using the forcings as inputs and treat the residual as being "natural variability" it will tend to over-estimate ECS, giving an upper bound estimate. The Abbot and Morahasy paper cites a similar paper by Loehle, but sadly does not also cite the comment paper (of which I was the lead author, note the corregendum). This is poor scholarship, and sadly Abbot and Morahasy have gone on to make many of the same basic mistakes (but with a more complicated model), which is a shame.
Rather than using the original datasets (many of which are freely available) the authors chose to digitise images of the datasets instead. This seems somewhat bizarre, and Gavin Schmidt points out via twitter that in at least one case the dataset has not been scaled or aligned correctly (andends at 1965, and so does not include the recent warming where anthropogenic contributions are most evident). It also transpires that Figures 5 and 9 are identical.
The paper says "However, superior fitting to the temperature proxies are obtained by using the sine wave components and composite as input data. This was established by comparing the spectral analysis composite method versus the ANN method for the training periods.". Evaluating performance on the training data is a classic error in the use of machine learning that people used to make all the time in the late 1980s and 90s, but is rarely seen today. If you have two nested models (one can be implemented as a special case of the other) of different complexities, then the more complex model will always have a lower training set error, if only because it has more capacity to memorize the random noise in the data, but that doesn't mean it is the more accurate model. For that you need out-of-sample comparisons, which are absent from the paper.
There is no handling of uncertainty in the model (for instance the periods of the cyclic components are not known exactly, cycles with slightly different periodicities will explain the observations almost as well), and likewise there will be uncertainty in the parameters of the neural net, but none of this is propagated through to give the uncertainty in the estimate of ECS. As seen in the comment on the Loehle paper, this can be substantial.
Table 13 seems to suggest that paleoclimate studies give lower ECS estimates than GCMs, which suggests a rather selective view of the paleoclimate studies, which generally indicate high ECS IIRC.
The study also has problems with too many degrees of researcher freedom (e.g. how was the particular subset of proxies chosen?) and there is a lot of (automated) exploration of model architectures and feature selection, which is often a recipe for over-fitting in model selection. It is also not clear why the observation should be a non-linear function of the cyclic variables (especially given that the cycles were obtained from the data by linear analysis).