DNNs are very good at learning features so are a natural fit if you have lots of training data. The main drawback is then you've lost the ability to work with small data. The other drawback is the computational cost of the maxent model on top of it. So you get both the best and the worst of both halves.
So:
input --DNN--> feature vector --minimax_entropy_optimisation--> maxent model --sampling/inference--> output
But minimax entropy learning is just learning a model that matches statistics of a set of features (filters). If you just want to draw a single sample from that model or the maximum likelihood point (e.g. you just want the answer to an IQ question) rather than doing complicated inference (in the DNN era, people have forgotten that "inference" is far more general than classification) you don't actually need the model at all, you can skip learning it, and go straight to computing a result with the desired statistics. This is around 100-1000x faster.
input --DNN--> feature vector --statistics_matching--> output
This is what I mostly did in my PhD work, and there's a famous paper doing this:
A Neural Algorithm of Artistic Style https://arxiv.org/abs/1505.07376
This has a sister paper which is even purer statistics-of-features-matching:
Texture Synthesis Using Convolutional Neural Networks https://arxiv.org/abs/1505.07376
However soon people worked out you can just train a DNN to do the statistics matching directly. Initially, with a fixed target style, and later taking the style as a second input. If the results are acceptable, this is vastly faster at inference time.
input --DNN--> output
Using them in this way would likely mean just replacing the final layer. The original papers on 'style transfer' did this, a DNN plus minimax entropy learning. has been done in texture and image models.