You need to get the energy input equal over the whole of your planet - two options spring to mind:
1) The albedo (~reflectance) of the planet could be finetuned so the further to the poles you get the lower the albedo.
This could be a result of the prevailing sediment at the different latitudes, and would not even have to be visible to the eye, if you decree the differences in reflectance to be entirely in the invisible wavelengths (e.g. IR).
Varying the local albedo via bodies of water would be neat (you would just have to vary the size and distribution of ponds, which explains itself quite neatly along the lines of latitude-dependent geology), but this changes the local ability to retain heat, as water has a quite exceptional heat capacity, so you'd have to introduce a mechanism that radiates heat faster in the regions with more water, to get the requisite drop in temperature over night - possibly another geological feature like craggyness of the ground (like heat vanes)?
Note that vegetation might interfere in your scheme by covering the ground - OR the ground cover has for some reason evolved to produce exactly those parameters. That would be neat, as plants can change shape, color and vapor-losses in small timescales, thus making a globally-eualized climate even more feasible by introducing an active control of sorts.
Having a reflective soil at the regions with maximal irradiance (equator), and a much more (heat-)radiation-absorbing soil in the lesser lighted areas at the poles would offset the difference in energy input, and would thus make for globally equal average temperature. To get the 45° C day/night cycle you have to decree a lack of 'insulation', i.e. no clouds, or at least globally uniform amounts of clouds.
2) Your planet is a cylinder (thus probably of non-natural origin) with it's axis orthogonal to the radius of it's orbit. Ignore the endcaps. This also guarantees globally equal irradiance.