The answer depends on the location of the desert. In the northern hemisphere January is in the winter. In the southern hemisphere it is summer. There are hot deserts in both.
The answer to your question depends upon the location of the particular desert.
The Simpson Desert of Australia is a hot desert.
The weather in Egypt in January is not as hot as you imagine. It is in the desert and in Africa but northern Egypt in January can be cold at night and it does rain.
Hot desert
The Gibson Desert is a hot desert.
A hot desert
The Arabian Desert is a hot desert.
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The Kalahari is a subtropical hot desert.
The Kalahari is classified as a hot sub tropical desert.
Yes, the Simpson Desert is a hot desert.
The Arabian Desert is a hot desert.