Here are two possibilities for frozen equator and warm poles.
One way is for the planet to spin rabidly and have a very oblate shape when molten and cool down and solidify while still oblate. The planet's rotation can later be slowed down to the desired rate by factors like the gravity of a moon or companion planet after the planet is solidified.
There are limits to how oblate a solid planet can be. But might be possible for a planet to be oblate enough for the planets various lawyers to be lot thinner at the poles than at the equator.
As we all know, the deeper a mine or other excavation goes, the hotter it gets from the internal heat of the Earth. If your planet is several times more oblate that the Earth, the difference in equatorial and polar radius will make the polar regions significantly closer to the source of the planet's internal heat.
The internal heat of the Earth comes about equally from primordial heat, heat left over from the formation of the Earth about 4,600,000,000 years ago, and radioactive decay. The cooling of Earth's primordial heat has slowed down over billions of years, and so Earth's primordial heat emission was greater in earlier ages. As radioactive isotopes in the crust and mantle decay, they produce less and less heat over time. Thus radioactive decay produced much more internal heat in earlier ages.
So a planet's surface will receive more heat from inside while it is younger than when it is older. But I don't think you can make your planet much younger than Earth to have more heat from the interior if you want it to have interesting features like a breathable atmosphere, advanced multi celled lifeforms on land, and/or intelligent native life, because all those appeared relatively recently on Earth on the geological time scale.
So you could increase the heat flow from the interior by making your planet somewhat more massive than Earth, giving it more primordial heat, and giving it a heavier original concentration of radioactive isotopes.
And you can add a third internal heat source that is not very important or Earth but could be very important on your world. Tidal heating.
The best way to get significant tidal heating on your planet would be to make it a giant natural satellite or moon of a gas giant planet. The gas giant planet would have to orbit in the habitable zone around the star, or possibly a bit farther out if the tidal heating is enough to compensate for the lower amount of sunlight received.
According to this article: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/1
A moon can orbit too close to a giant planet and get heated up so much by tidal heating that the moon will be too hot, and/or, like Io, will have too much volcanic activity, to be habitable. But if a moon orbits too far from the gas giant planet it will not be bound tightly enough to the planet and it can easily be perturbed by other astronomical objects and escape from orbit around the planet.
Thus a gas giant will have a habitable zone in which a very large and potentially habitable moon can orbit and not be overheated and not be lost into space. And the size of that habitable zone can be calculated from the mass of the gas giant planet. The length of the moon's monthly orbit around the planet can be calculated from the mass of the planet and the distance the moon orbits around the planet.
A potentially habitable giant moon of a gas giant planet will have its rotation period slowed down to match its orbital period. It will keep one side always facing the planet and one side always facing away from the planet. but it will continue to rotate relative to its sun with a day equal in length to the orbital period around the planet.
Of course if the habitable moon has a day that is too long it will heat up too much in daylight and cool down too much in darkness, so you will want the moon to have a day that is not too long, which means that its orbital period around the gas giant planet must not be too long.
And you will want the planetary sized moon to receive less heat and light from its sun than Earth does, so the moon would not be habitable except for the extra heat coming from the interior due to intense tidal heating.
If the planet sized moon is significantly more oblate than Earth, the polar regions, closer to the core, might receive enough heat from inside to have liquid water, while the equatorial regions might been too cold for liquid water. Thus water from the polar regions would evaporate and be carried by winds to the equatorial regions were with would become ice in the glaciers. Many of the glaciers would flow to the north and the south and would melt when they reached warm enough regions, thus returning water to the warm polar regions.
And possibly your planet/moon could have a tall equatorial ridge like Iapetus. Glaciers could form on the equatorial ridge and flow down it to warmer latitudes melt.
And to make one of the low polar regions even lower, perhaps millions of years ago a giant asteroid struck one of the polar regions and exploded and formed a vast concentric impact basin centered near that pole. And perhaps material ejected from that impact came raining down on the equator to form the equatorial ridge. This would make one of the polar regions lower and maybe warmer than the other.
If a hypothetical planet rotates around an axis that is perpendicular to the plane of the planet's orbit around its star, the planet has an axial tilt or obliquity of zero degrees. If, on the other hand, the hypothetical planet rotates around an axis that is in the plane of the planet's orbit around the star, the planet would have an axial tilt of 90 degrees.
The real planets in our solar system have axial tilts varying from 0.03 degrees (Mercury) to 82.23 degrees (Uranus).
That means that during part of the Uranian year (84.0205 Earth years long) the north pole of Uranus will be pointed toward the sun and in constant light for years at a time and the south pole of Uranus will be pointed away from the Sun and in constant darkness for years at a time.
And then, 42.01025 Earth years later, the south pole of Uranus will be pointed toward the sun and in constant light for years at a time and the north pole of Uranus will be pointed away from the Sun and in constant darkness for years at a time.
And halfway between those two periods, 21.005125 Earth years before and after, Uranus would be positioned where sunlight would fall on it almost parallel to the equator of Uranus. Since Uranus has a day about 0.71833 Earth days long, every region of the planet would have about 8.619 hours of light followed by about 8.619 hours of darkness.
So if your planet has an axial tilt close to 90 degrees then there will be periods during its year when one pole is pointed at its star and has constant light. Thus the pole will heat up very much. The farther away from the pole one got, the greater the angle that the sunlight would be coming down at the less it would heat the air and the ground. At the planet's equator the sunlight would be coming down almost parallel to the ground and wouldn't heat up the ground much. And the other side of the planet would be in constant darkness and would get colder and colder.
Water vapor from the pole facing the star would flow in winds toward the opposite side of the planet, and would freeze out at the equator and/or the opposite side the planet.
And half a planetary year later, the other pole would be in constant light and heat up. The glaciers forming there would melt and some of the water vapor would be carried away by winds and freeze out on the equator and on the side now in darkness.
And halfway in between those two extremes, the planet would get starlight coming almost straight down at the equator which would warm up a lot, with much of the ice evaporating, and coming down at a great slant at the poles which would not be heated up much. The planet would have short days and nights everywhere.
And in the intermediate periods the planet would have intermediate climate.
It is possible (but not certain) that the poles would heat up enough when they pointed at the star to have tropical weather during their warm seasons, but the equator would never get warm enough to completely melt the glaciers there. Thus the equatorial glaciers would get bigger and bigger until the equatorial regions were covered in ice sheets.
The disadvantage of this is that the polar regions would have cold seasons so they both would have rather temperate climates, not tropical, with alternating winters and summers, instead of constant summer.
One way to minimize this, as well as reduce unbearable temperature extremes, would be to make the planet have a very short year due to a small orbit close to a dim star. Thus the polar regions, especially if they have plenty of water that retains heat, might not cool off too much between "summers".
I would guesstimate that it would be good for your planet to have a year shorter than 84.0205 Earth days (2,016.492 hours) but longer than 84.0205 hours (3.5008541 Earth days). Wikipedia's list of potentially habitable exoplanets list planets with years ranging from 4.05 Earth days to 384.8 Earth days, with 10 planets having years in the the range I guessed at.