Just bought a house and am finding the hot water pipes are smaller than the cold water pipes. Why?
Hot water delivery times are highly dependent on the size of the pipe. It will take more than twice as long for hot water to reach the fixture travelling through a 1/2" pipe than a 3/8" pipe (because there is roughly twice as much cold water sitting in the pipe), and more than twice as long through a 3/4" pipe compared to a 1/2" pipe. Thus, using the smallest supply lines possible for hot water plumbing to fixtures will supply hot water much faster, saving water and energy, and making occupants happy.
With cold water supply lines, there's no similar issue. Instead, our only concerns are about supplying enough water for good flow rates, and stable pressure (so output doesn't noticeably fluctuate when someone flushes a toilet, for example). So for the same fixture where a 3/8" hot water supply line makes sense, it also makes sense to use a 1/2" cold water supply line.
For part of the system this may make perfect sense - i.e. the main cold feed line into the house, which is carrying all the cold and "will be heated to become hot" water at the same time, so it should be larger than the individual feed lines to fixtures.
If it's extended all the way to individual fixtures with larger cold water and smaller hot water lines, it makes less sense and is tending to make things unbalanced, though again, it can make sense for a "bathroom set" where the hot line is feeding the tub/shower and sink, but the cold line is also feeding the toilet. A larger cold line there might make "flush-scald" where the cold water pressure drops and the shower is suddenly hot less likely, or less severe.
Why? You'd have to dig up whoever plumbed it and ask...many things in houses don't make a lot of sense unless you are the guy who chose to do it that way, for whatever inscrutable-after-the-fact reason.
It is completely normal to have your cold and hot water lines be different sizes. Your boiler or your water heater probably has a smaller output size than the main coming in.
Even though the water pipes differ for the most part, the size of the water supply pipes to your fixtures would be the same for both hot and cold lines. Standard sinks and tubs would use 1/2" supply lines, and toilets use a single 3/8" line.
As long as there is enough volume and pressure in your system, you shouldn't have any issues.