Yes, even on earth, there is a depth below surface where temperature is the average annual temperature of the above-earth (water, actually, in most of the earth so not much variation). On land, this mean annual temperature is found at a few to several meters below ground surface: the ground between surface and that constant temperature will vary over the seasons (whips back and forth between summer high and winter low, attenuating (going to zero change) with depth. How deep the whipsaw variations extend depends on the intensity of the change at surface.

Caves and even relatively shallow storage buildings dug into the ground rely on this stability of temperature at depth, so the air in such places tends to be pretty much the same temperature all year round, and until you go very deep like with some mines (where heat from below is enough to raise temperatures; we are deep enough to be well below that depth of mean average temperature so heat from below is on its way to the surface), that temperature is the mean annual temperature of the location.

The idea is that earth surface is at the temperature where solar heating is balanced by black body radiation. Clearly, when the earth was very hot in its youth, black body radiation (emission of "light" energy based on temperature) was much higher than solar heating, so the surface of the earth rapidly cooled due to excess loss of heat to space (much more heat lost to space than gained from space), but the heat loss from the very hot early earth was rapid and the (almost) steady-state balance that now exists came to dominate billions of years ago. Not quite steady state, because the heat flux from below is not zero, so there is always slightly more heat being lost to space beyond the amount that is received from space, but the difference is very small now. Loss of internal heat is very slow. It is a factor, but a tiny one.

Now, what we see is the rate of heat migration from inside to surface has "mean surface temperature" as the lower limit for the geothermal gradient. Locally, like when there are massive magma intrusions to shallow depth, there can be a temporary disruption of the balance and heat loss in that region can be measurably higher than average annual solar heating, but it lasts briefly only, like a million years time frame (time frame depends a lot on how active hydrothermal fluid convection is, because convection is way faster than conduction).

When the interior of a planet falls to the mean average surface temperature, there is no more migration of heat from the interior. The entire planet would be kept at that mean surface temperature. Planets are so large, and heat flow by conduction through rock is so slow (absent convection by circulating or migrating fluids and the occasional rising blob of magma) that no planetary bodies that we know about are that cold, yet.

Small bodies like asteroids out in space have very cold internal temperatures but are not at the temperature of deep space because they do get warmed slightly by the sun. On earth, where seasons happen, the summer is a period where heat received by the sun is more than is lost to space, but heat during winter is less than is lost to space. The mean annual temperature is the temperature where those shorter-term losses and gains get balanced to no change.

I suppose even asteroids and comets have "seasonal" variations, even if the seasons are many years long. Clearly, a comet near the sun that is degassing is in what can only be seen as a form of "summer", even if the seasons are imposed from orbital variations rather than tilted axis of rotation. Earth does also have some orbit-dependent heating change, but the orbit is almost circular so not a huge different. It does matter though (see Milankovitch cycles).

Also, the presence of hydrosphere and atmosphere are important in determining what that "mean annual surface temperature" will be. Greenhouse effects, of a sort. This is why Venus is way hotter at the surface than it ought to be based on simple solar flux considerations. So much of its radiant heat loss to space gets trapped by the atmosphere that an important portion migrates back to surface if the surface cools down, so the stable or steady state temperature at surface is higher than it would be in absence of atmosphere. It cannot radiate heat to space at the rate that its surface temperature would have it. This is also true on earth but to a much smaller extent.

The basic problem is that heat cannot leave where it is now unless there is somewhere cooler for it to move into. So, inside the earth (or any planet, really) the internal heat is simply unable to leave except very slowly.