A method of cooling a seal seat (72; 172), the method comprising: delivering a fluid coolant to a first portion (80; 180) of the seal seat (72; 172), wherein the seal seat (72; 172) comprises: an annular body (74; 174) disposed about a centerline axis (C A), the annular body (74; 174) extending from an upstream end of the annular body (74; 174) to a downstream end of the annular body (74; 174); and an extension connected to and extending radially outward from a first axial end of the annular body (74; 174), wherein the extension comprises: the first portion (80; 180) connected at an inner radial extent to the annular body (74; 174) and defined by an upstream wall face (82; 182) and a downstream wall face (88; 188) axially opposite the upstream wall face (82; 182), wherein the first portion (80; 180) comprises a fin (84; 84A; 84B; 184) at the upstream wall face (82; 182) and extending axially away and upstream from the downstream wall face (88; 188); a second portion (90; 190) connected to an outer radial extent of the first portion (80; 180), wherein the second portion (90; 190) extends substantially parallel to the annular body (74; 174), and axially upstream from the first portion (80; 180); and a third portion (94; 194) connected to the second portion (90; 190) at an end of the second portion (90; 190) that is distal from the first portion (80; 180), wherein the third portion (94; 194) extends substantially parallel to the first portion (80; 180); forming a film of the fluid coolant on the upstream wall face (82; 182) of the first portion (80; 180) of the extension; flowing the film of the fluid coolant radially outward and across the upstream wall face (82; 182) and across the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182); and transferring thermal energy from the first portion (80; 180) of the extension to the film of the fluid coolant via the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182). an annular body (74; 174) disposed about a centerline axis (C A), the annular body (74; 174) extending from an upstream end of the annular body (74; 174) to a downstream end of the annular body (74; 174); and an extension connected to and extending radially outward from a first axial end of the annular body (74; 174), wherein the extension comprises: the first portion (80; 180) connected at an inner radial extent to the annular body (74; 174) and defined by an upstream wall face (82; 182) and a downstream wall face (88; 188) axially opposite the upstream wall face (82; 182), wherein the first portion (80; 180) comprises a fin (84; 84A; 84B; 184) at the upstream wall face (82; 182) and extending axially away and upstream from the downstream wall face (88; 188); a second portion (90; 190) connected to an outer radial extent of the first portion (80; 180), wherein the second portion (90; 190) extends substantially parallel to the annular body (74; 174), and axially upstream from the first portion (80; 180); and a third portion (94; 194) connected to the second portion (90; 190) at an end of the second portion (90; 190) that is distal from the first portion (80; 180), wherein the third portion (94; 194) extends substantially parallel to the first portion (80; 180); forming a film of the fluid coolant on the upstream wall face (82; 182) of the first portion (80; 180) of the extension; flowing the film of the fluid coolant radially outward and across the upstream wall face (82; 182) and across the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182); and transferring thermal energy from the first portion (80; 180) of the extension to the film of the fluid coolant via the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182). the first portion (80; 180) connected at an inner radial extent to the annular body (74; 174) and defined by an upstream wall face (82; 182) and a downstream wall face (88; 188) axially opposite the upstream wall face (82; 182), wherein the first portion (80; 180) comprises a fin (84; 84A; 84B; 184) at the upstream wall face (82; 182) and extending axially away and upstream from the downstream wall face (88; 188); a second portion (90; 190) connected to an outer radial extent of the first portion (80; 180), wherein the second portion (90; 190) extends substantially parallel to the annular body (74; 174), and axially upstream from the first portion (80; 180); and a third portion (94; 194) connected to the second portion (90; 190) at an end of the second portion (90; 190) that is distal from the first portion (80; 180), wherein the third portion (94; 194) extends substantially parallel to the first portion (80; 180); forming a film of the fluid coolant on the upstream wall face (82; 182) of the first portion (80; 180) of the extension; flowing the film of the fluid coolant radially outward and across the upstream wall face (82; 182) and across the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182); and transferring thermal energy from the first portion (80; 180) of the extension to the film of the fluid coolant via the fin (84; 84A; 84B; 184) of the upstream wall face (82; 182).