But we effortlessly move through time, yet to move through space does take effort, ie the expenditure of energy. Why?

On our current understanding of physics there are deep connections between some fundamental concepts. One of these is that every physical symmetry leads to a conservation law.

So the symmetry of

*time-translation *i,e. we assume that a closed system experiment will produce the same results at different times, keeping other factors the same, implies the conservation of energy.

And the symmetry of

*spatial-translation *i.e. moving a closed system experiment in space, keeping other factors the same, does not affect the result of the experiment and implies the conservation of momentum.

See Noether's theorem for details. It's all Lagrangians and differential calculus.

If you know anything about Heisenberg's uncertainty principle this 'checks out'! Namely where the fundamental limit in accuracy is linked between momentum and position, and also there is a similar relationship between energy and time.

Add these conservation laws to our fundamental understanding of forces (which are essentially built into Noether's theorem, anyway) led to the basic physics we observe. So to answer your question. If there is an object at rest and you move it to another position you have to apply a force. The object reacts to this force 'obeying' both the conservation of momentum and energy, the initial force applied being your 'expenditure of energy'.

**However***, *you'll note for the cosmos as a whole, if we assume the big bang theory, time had a beginning thus time-invariance is fundamentally broken. (the above conservation laws apply to closed systems inside the universe.) Hence for the cosmos there is no conservation of energy - for example as our universe expands the amount of 'dark energy' increases, the universe doesn't need to borrow energy from elsewhere.

Anyway

Finally, I think I would disagree with your statement that we 'move effortlessly through time'. Only an object that does not change state and is not influenced by any external forces will demonstrate 'effortless chrono-mechanics'. But such objects are 1) uninteresting 2) Essentially not present in our current universe - note that everything we see is definitely influenced at the very least by the gravitational force - local bodies such as planets, the Milky Way, galaxy clusters etc.

Thus assuming we believe that there is a relationship between our consciousness and the physical state of our brains, just to think we move electrons and other molecules around our brain, causing our state to change. Thus to be us we must expend energy to move physical objects. Progressing through time needs energy and interactions to change states to give rise what we experience.