A vehicle air compressor apparatus for a heavy vehicle air braking system, the vehicle air compressor apparatus comprising: a compressor crankcase assembly; a compressor cylinder head assembly disposed on the crankcase assembly and including (i) an air inlet port through which air can be received for compression within the crankcase and cylinder head assemblies, (ii) a discharge port through which compressed air can be delivered from the cylinder head assembly, (iii) a first intermediate port through which compressed air can pass, and (iv) a second intermediate port through which cooled compressed air can pass; and a discharge air cooling jumper (400, 500, 600, 700) connected externally of the cylinder head assembly between the first and second intermediate ports and arranged to cool compressed air passing from the first intermediate port through a discharge air path of the jumper to the second intermediate port and thereby to provide cooler compressed air to be delivered from the cylinder head assembly through the discharge port; wherein (i) the discharge air cooling jumper includes a first end portion defining a first end of the discharge air path and a second end portion defining second end of the discharge air path, and (ii) the discharge air path extends in a substantial U-shape between the first and second ends of the discharge air path; and wherein (i) the first end portion of the jumper defines a first jumper bearing surface which faces a first cylinder head bearing surface in the vicinity of the first intermediate port of the cylinder head assembly, and (ii) the second end portion of the jumper defines a second jumper bearing surface which faces a second cylinder head bearing surface in the vicinity of the second intermediate port of the cylinder head assembly. a compressor crankcase assembly; a compressor cylinder head assembly disposed on the crankcase assembly and including (i) an air inlet port through which air can be received for compression within the crankcase and cylinder head assemblies, (ii) a discharge port through which compressed air can be delivered from the cylinder head assembly, (iii) a first intermediate port through which compressed air can pass, and (iv) a second intermediate port through which cooled compressed air can pass; and a discharge air cooling jumper (400, 500, 600, 700) connected externally of the cylinder head assembly between the first and second intermediate ports and arranged to cool compressed air passing from the first intermediate port through a discharge air path of the jumper to the second intermediate port and thereby to provide cooler compressed air to be delivered from the cylinder head assembly through the discharge port; wherein (i) the discharge air cooling jumper includes a first end portion defining a first end of the discharge air path and a second end portion defining second end of the discharge air path, and (ii) the discharge air path extends in a substantial U-shape between the first and second ends of the discharge air path; and wherein (i) the first end portion of the jumper defines a first jumper bearing surface which faces a first cylinder head bearing surface in the vicinity of the first intermediate port of the cylinder head assembly, and (ii) the second end portion of the jumper defines a second jumper bearing surface which faces a second cylinder head bearing surface in the vicinity of the second intermediate port of the cylinder head assembly.