The　finite-time　thermodynamic　performance　has　been　studied　of　an　endoreversible　air　heat-pump　with　constant-temperature　heat-reservoirs.　The　heating　load,　the　coefficient　of　performance　(COP),　and　the　heating-load　density,　i.e.　the　ratio　of　heating　load　to　the　maximum　specific　volume　in　the　cycle,　are　the　optimization　objectives.　The　analytical　formulae　relating　the　heating　load　and　pressure-ratio,　between　the　COP　and　pressure-ratio,　as　well　as　between　the　heating-load　density　and　pressure-ratio　are　derived　assuming　heat　resistance　losses　occur　in　the　hot-　and　cold-side　heat-exchangers.　The　influences　of　the　effectiveness　of　the　heat-exchangers　and　the　heat-reservoir　temperature-ratio　on　the　heating　load,　the　COP　and　the　heating-load　density　are　analyzed.　The　cycle　performance　optimizations　are　performed　by　searching　the　optimal　distribution　of　heat　conductance　of　the　hot-　and　cold-side　heat-exchangers　for　the　fixed　total　heat-exchanger　inventory.　The　influences　of　some　design　parameters,　including　heat-capacity　rate　of　the　working　fluid,　heat-reservoir　temperature-ratio　and　heat-exchanger　inventory　on　the　optimal　distribution　of　heat　conductance,　the　maximum　heating　load　and　the　maximum　heating-load　density　are　indicated　by　numerical　examples.　The　different　results　obtained　from　the　heating-load　optimization　and　the　heating-load　density　optimization　are　shown.　The　air　heat-pump　design,　with　heat-loading　density　optimization,　leads　to　smaller　size　equipment.　(c)　2007　Elsevier　Ltd.　All　rights　reserved.